Is there a name for the philosophical position that states that moral responsibility is compatible with the lack of free will?
The standard label for this position is 'soft determinism' or 'compatibilism'.
Compatibilism is, roughly, the position that the forms of free will most people clearly have to some degree, such as the ability to deliberate and do as they wish [without interference from others: GT], suffice to meet the requirements of morality and personal life insofar as they are affected by the issue of free will. In particular, the compatibilist rejects the idea that some sort of 'metaphysical' or 'libertarian' notion of free will, such as would be negated by a completely deterministic ontology, is necessary in order to have moral responsibility. Hence, the term 'compatibilism': the compatibilist insists that free will, moral responsibility, and their concomitant notions are compatible with determinism (or with the absence of libertarian free will). For example, the compatibilist would claim that most people in the West choose a career with some measure of freedom, and are morally responsible for this choice, although it follows from their desires and beliefs. Lack of relevant freedom would result only from atypical causes eliminating or severely curtailing control (such as pathological compulsion or external coercion). It is important to stress that compatibilism is not utilitarian or consequentialist but maintains contact with the traditional paradigm requiring control for moral responsibility, and moral responsibility for blameworthiness and desert. Compatibilists maintain that the traditional paradigm can be sustained even in a deterministic world, and does not require libertarian free will. On the compatibilist level of deliberating, choosing, and acting, most people are basically free, such matters are within their control, and it is this that matters.
(Saul Smilansky, 'Compatibilism: The Argument from Shallowness', Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 115, No. 3 (Sep., 2003), pp. 257-282: 259-60.)
Put the point like this. Suppose determinism is true and that my choices and intentions are all part of a purely deterministic world. I cannot choose other than I so, and I cannot form intentions other than those I do. There is a strong inclination to say that in such a world I am not a free agent and lack moral responsibility.
But the compatibilist redefines the conditions of moral responsibility, and specifically the freedom requisite for moral responsibility. For the compatibilist I am free if in acting I am exempt from constraint. If I choose to buy a packet of gum and intend to do so, then my corresponding action is free as long as no-one inteferes. I am not free if you block my path or deprive me of my money. Hence the compatibilist slogan, 'Freedom is not opposed to causation but to constraint'. My actions may be fully causally determined but I am 'free' if I act without interference and am in this sense the author of actions.
How, though, is my moral responsibility preserved? The compatibilist holds that as long as I remain responsive to moral criticism or appraisal, I remain morally responsible for my actions. If you can alter my actions (or my attitude towards them) by making moral judgements on them, then my amenability to this kind of moral control is sufficient to satisfy the conditions for moral responsibility. This is so, even though causation is operative throughout. Whether this yields an adequate account of moral responsibility, it is in line with the compatibilist's account of freedom. Moral responsibilty is not opposed to causation but to constraint. If my actions are subject to constraint, as when you force me to act, then I am not morally responsible for them.
I am not aware of any specific label that identifies the position that not only is free will compatible with determinism ('compatibilism' as minimally characterised) but that determinism is compatible with moral responsibility. But it is implicit in compatibilism as outlined in the opening quotation, and part of the point of the position, that the freedom presupposed to moral responsibility is unaffected by determinism. Just as I said above: 'The compatibilist holds that as long as I remain [deterministically] responsive to moral criticism or appraisal, I remain morally responsible for my actions. If you can alter my actions (or my attitude towards them) by making moral judgements on them, then my [deterministic] amenability to this kind of moral control is sufficient to satisfy the conditions for moral responsibility.'
The standard critique, which I share, is that the compatibilist's redefinition of terms, however reasonable, alters but does not eliminate the issue between hard determinists and libertarians. On that issue you will need to do some research.