I agree with Ahmed. By far, the best answer would be:
e: They had their lawyer change their will.
But since this is not offered as an option, the only possible answer is:
c: They had their lawyer changing their will.
As Ahmed says, this is grammatically correct. However, as a native speaker of English, it sounds very unnatural to my ears. It is not always possible to say why such a thing might be, but in this case I think I can make a fairly good guess.
Consider these sentences:
e': I had George guard the back door.
c': I had George guarding the back door.
Both of these sentences are grammatically correct and sound natural. Because "to guard" cannot be a single instantaneous action, the meaning of (e') implies continuous action, which means that any difference in meaning that may exist between (e') and (c') is rather subtle.
e'1: Marla came in while George guarded the back door.
c'1: Marla came in while George was guarding the back door.
However, because "to change" does tend to indicate a single simple action in the simple past and simple present, there is a significant difference in meaning between (e) and (c). In my mind, the meaning of (c) would be much better rendered as:
c2: They were having their lawyer change their will.
I look forward to feedback from my fellow members, especially David, Gustavo, and Ahmed, and of course Yama, who brought this question up in the first place.
One more thing:
According to what I know, " have " in the causative form is followed by an "infinitive".
This is true, within certain limits. I think that many exceptions have been demonstrated here already, where forms other than the infinitive will work in such a construct. Even so, we have to make a distinction between the simple infinitive (e) and the "to + V" form (b).
(b) would have been the best answer had it been offered as the simple infinitive (without the particle "to").