In books many material objects were transfigured into animals or just another objects. So can a wizard use transfiguration spell with another wizards' wands?
There is little reason to conclude that it can't be done, though it may be so difficult as to be effectively impossible.
In the books, transfiguration isn't really shown to have fundamental limits so much as practical limits. Many varieties of transfiguration are described as difficult, but are clearly possible for children in school and appear casually possible for capable adults:
- Viktor Krum partially transfigures himself into a shark so that he can spend more time underwater
- Cedric Diggory transfigures a rock into a terrier as a distraction, which is notably comment-worthy in Ron Weasley's opinion (a pretty cool bit of transfiguration), but not mind-blowing
- "Professor Moody" transfigures Draco Malfoy into a ferret as an offhanded punishment. He is scolded over this by Professor McGonagall, but she mentions nothing about danger being the reason
- Sirius Black, James Potter, James Lupin, and Peter Pettigrew manage to become animagi through self-study during their spare time while in school (described by Hermione as being extremely dangerous, that danger being great enough to prompt some related laws)
There are some arbitrary limits on what magic can do, such as Golpalott's Third Law indicating that a witch or wizard cannot simply conjure food out of nothing, but I don't recall any similar limitation being described with respect to transfiguration.
The only things we see which are fundamentally resistant to magical manipulation seem to be fundamentally magical in themselves-- it's hard to affect a dragon, acromantula, or giant with spells of any type. It's not clear if this limitation can be overcome with more magical might or any sort of cleverness (for example, you can cast a spell at a dragon's eyes, effective because the resistance seems to be in their scales).
This resistance is very idiosyncratic: it's hard to bespell a giant, or even a half-giant, but it seems you can affect a half-Veela just fine. Dementors are quite susceptible to a particular charm, but other spells must not be effective against them (otherwise, there would be little value in mastering the Patronus charm versus just blasting stunning spells at them).
And that presents the big reason that it might not be possible to transfigure a wand: each magic wand has as its core a magically active ingredient drawn from a magical creature. Are those pieces of those creatures resistant to magic in any way? We really don't know, as none of them are really described beyond being components in wands.
I can think of only one example which suggests that wands cannot be transfigured: in Chamber of Secrets, Ron's wand is broken and malfunctions in unpredictable ways as a result. Magically mending damaged objects is pretty common in the books, and so that Ron's wand isn't repaired magically (despite the ongoing problems the broken wand causes) may suggest that wands are to some degree magic-resistant. It is, of course, just as possible that one needs to know a lot about wands to repair one by any means, and as Ollivander describes there are relatively few people who know anything about wandlore.
We don't know anything about any hard limits on what transfiguration can affect, or what it can do to things it can affect. We have some reasons to suspect it might be difficult or impossible based on what wands are made of, but nothing like definitive evidence in either direction.
It's a bit hard to get a handle on what magic can and cannot do in the Harry Potter series. Very little about what magic is or how it operates is described, so we mostly only have demonstrated examples on which we can base some deductions.
With that in mind, witches and wizards seem to spend relatively little time and effort using transfigurations to solve their various problems, particularly with other magically-capable individuals. The implication of this is that doing so is either impossible or isn't worth the effort, with no good way to judge between them.
This is a recurring theme in the franchise, as the things we see magic is capable of accomplishing suggest that fully-educated witches and wizards should probably be doing a lot more magic than they seem to do, but there is no way to tell if this is an intended piece of the setting or oversight in writing.
Ultimately, from what we actually observe in the franchise, whether or not it is literally possible seems to be secondary to whether or not it's worth doing to transfigure a wand into something else. Expelliarmus seems like the most natural choice to deprive a witch or wizard of their wand, and appears to be far easier to do than a difficult transfiguration.