After a bit of research, it seems clear to me that - except in exceedingly rare, if not totally nonexistent circumstances - the laity ought not take communion at an SSPX chapel, or otherwise formally participate in the illicit liturgy. That the priests themselves ought not administer or say the illicit Masses is given in that they are illicit, this fact being confirmed by multiple sources, one binding, one traditionalist.
The argument itself is straightforward: The Church has spoken specifically to this matter in an interpretive fashion, and authoritative interpretation have the same force as the law itself, if not universally, at least in respect to particular cases. This is a particular case that has been spoken on, however, so the Church’s interpretation of the law is binding.
From STATUS OF SOCIETY OF ST PIUS X MASSES, Commission Ecclesia Dei:
"2. The Masses [SSPX] celebrate[s] are also valid, but it is considered morally illicit for the faithful to participate in these Masses unless they are physically or morally impeded from participating in a Mass celebrated by a Catholic priest in good standing (cf. Code of Canon Law, canon 844.2). The fact of not being able to assist at the celebration of the so-called "Tridentine" Mass is not considered a sufficient motive for attending such Masses." (Emphasis added)
Why the interpretation is necessarily binding:
Can. 16 §1. The legislator authentically interprets laws as does the one to whom the same legislator has entrusted the power of authentically interpreting.
Nor is this solution repugnant to all notable traditionalist sources:
Fr. Z's take on it:
"My solution would be that you could attend even an illicit Mass, though I would hope with some restraint, but that you should not receive Communion."
The referenced canon:
"§2. Whenever necessity requires it or true spiritual advantage suggests it, and provided that danger of error or of indifferentism is avoided, the Christian faithful for whom it is physically or morally impossible to approach a Catholic minister are permitted to receive the sacraments of penance, Eucharist, and anointing of the sick from non-Catholic ministers in whose Churches these sacraments are valid."
Traditionalists will argue that a true spiritual advantage suggests it. Note, however, that it has been clarified with binding authority that "the fact of not being able to assist at the celebration of the so-called 'Tridentine' Mass is not considered a sufficient motive for attending such Masses."
In more general terms, for any action to be moral, the object, circumstances and intent must all be acceptable. (Moral Theology 101) From this arises the generally accepted fact that anyone can attend even a protestant service so long as he does not formally participate in it (the object of the action), so long as he does not cause scandal (the circumstances of the action), and so long as he does so for a genuinely good reason (the intention of the action), and the same can be said of attending an SSPX Mass. Protestantism is further removed, so it makes sense that what we can do in regards to their services, we can also do in regards to SSPX's Masses. That the intention can be good need not be visited.
For the object of the action to be morally acceptable, either he must be incapable of morally or physically participating in a licit Mass, or else he cannot formally participate in an SSPX Mass. In that taking communion does constitute formal participation (Lest there be any doubt, this is what the cited canon referred to specifically, for the sacrament itself is illicit), a person cannot licitly receive the sacrament unless it is morally or physically impossible to receive it licitly elsewhere.
In regards to the circumstances, scandal must be avoided, schismatic attitudes and indifferentism must not be allowed to take root. The latter especially seems that it would be difficult to satisfy. When something is done often enough out of preference, the distaste for what it lacks (in this case licitness) is numbed, leading almost necessarily to an attitude of indifferentism in regards to whether a sacrament or liturgy is licit.
Is it not fitting after all, that the Sacrament called "Communion" should be received in full union with Rome?
There are other things, however, which are different in different places and countries [...] In regard to these and all other variable observances which may be met anywhere, one is at liberty to comply with them or not as he chooses; and there is no better rule for the wise and serious Christian in this matter, than to conform to the practice which he finds prevailing in the Church to which it may be his lot to come. For such a custom, if it is clearly not contrary to the faith nor to sound morality, is to be held as a thing indifferent, and ought to be observed for the sake of fellowship with those among whom we live. Augustine, Book I of Replies to Questions of Januarius, Par 2.
This is a counter-intuitive teaching. One would think that it would be more important to give God the highest form of worship, regardless of what others are doing, than to fit in with everyone else. St. Augustine does not seem to be of this opinion, however. In fact, he followed this passage immediately with the example that a person ought not to fast on Saturdays when in a community whose custom is not to fast on Saturdays. To fast is good by its very nature. Voluntarily accepting hunger in order to be more wholly detached from worldly pleasures and more totally given to God will bring us closer to God. This counter-intuitive teaching is driven further by Augustine in Par 5, in which he speaks against the condemnation of one place’s custom as a response to finding an apparently better practice elsewhere, saying that such condemnation "would show a childish weakness of judgment against which we should guard ourselves, and which we must bear with in others, but correct in all who are under our influence."
Although this clearly does mean that whether we follow a custom should chiefly be determined by what the community is doing, this is not a license to relativism. One practice may very well be better than another, but it is more important that we are humbly united in Christ under our bishops than that our customs are perfect. St. Augustine is not saying that no custom is better than another, but is saying that it is more important for a community to be united in a good worship than it is for an individual to perform the best worship. St. Augustine very clearly viewed acceptance of a custom by the universal Church to be authoritative proof that it is good (pars. 1, 5, 6, 8), going so far as to say, "If the universal Church follows any one of these methods, there is no room for doubt as to our duty; for it would be the height of arrogant madness to discuss whether or not we should comply with it." (Par 6)
We can learn from St. Augustine our proper place as the laity in regards to disciplines of the Church. We ought to abide by local customs, even when another custom may be better, for the sake of unity. Acceptance by the universal Church authoritatively proves a custom to be good, to such an extent that St. Augustine referred to mere discussion of whether we ought to comply with it as "the height of arrogant madness". We ought to accept the disciplines of the Church then with docile humility, only going beyond the customs when doing so would not be disruptive.