It's an auxiliary verb, so I'm not sure whether or not it can conjugate. For example, is it correct to end a sentence with「。。。が欲しがります」？ Also, if you, for example, wanted to say that someone wants to want something, would「[誰か]は[何か]が欲しがりたいです」work?
Yes, it can conjugate just like a normal godan verb.
- 欲しがります (-ません, -ました, ...)
(This is rare as the "let's want" sense, but can be used as part of the (よ)うと/(よ)うが construction, e.g., 欲しがろうが欲しがるまいが与えられる "is given regardless of whether one wants it or not")
But it has many semantic restrictions, so please read this before using them.
When do you ever say "I want to want it" in English in the first place? 欲しがりたい may be usable in a very limited context where the way you show your desire to someone is important, but it's probably best to forget something tricky like this until you reach the near-native level.