Simply put, do pets have the chance to crit when they attack? If so, would the critical damage be improved by Infinity Edge if bought?
I do not think so due to the fact that it is its own entity that does not share your stats, with the exception of shaco's clone, from what I have read it appears to duplicate all on hit effects including life steal and crits
Okay I have done some research now manually. I have gone into custom games and built up a full crit build, then when I have roughly 75% crit chance I used the pet only to see if it dealt any crits. Very few of the 'pets' gains crit from the owners critchance. The real question is not whenever or not the pets crit, but if they GAIN crit from the champion.
Malzahar's voidlings doesn't get crit from Malzahar, they gain AD from him though.
Shaco's boxes doesn't gain crit from Shaco, but his clone does. Infinity Edge works as well.
Annie's Tibbers, doesn't gain crit, he scales in power through Annie getting more AP.
Elise spiders doesn't gain crit, they gain power through Elise getting more AP.
Yorick's ghouls doesn't gain crit, they gain from AD and Health. Yorick's ghost doesn't gain crit from Yorick, but from whoever he clones it gains same items as, and thus gain crit chance that way.
Mordekaiser's ghost crit chance is only influenced by victims items, not by Mordekaiser. Even if Mordekaiser gains crit chance that doesn't transfer. The ghost gets stronger with Morde gaining AD, AP and Health though.
Zyra's plants doesn't gain increased crit chance with Zyra getting it. They get stronger with her getting AP.
LeBlanc's clone doesn't deal damage and cannot crit, the clone does utilize LeBlanc's on-hit effects though, and deal damage that way.
I'm going to say no. As defined: