It seems possible
I've reached out to Elio and Linda, the co-authors of The World of Ice and Fire and whilst their answer isn't canon and not really semi-canon it generally holds more weight than others. They have said valar is likely to come from Valyria rather than Tolkien or anything else.
@CarrotLethal: Hello! I was just wondering do you know if the word "valar" was inspired by the word "Valyrian" when GRRM created it or was it a case of "it sounds right for Valyrian" or is it a reference to the Valar from LOTR or anything else? Thanks
@westerosorg: I am pretty sure "valar" comes from "Valyria" rather than from Tolkien.
Note: I am not a linguist and am essentially in the same boat as GRRM where if I ever create a language I'd go for nice sounding words. Therefore, my answer is mainly pieced together from what GRRM has said in the past and what I know of the very little information on the language.
See my research into this below:
I've done some research into the languages now, and in the past, and there is generally very little information on the ASOIAF canon languages because George himself has very little information on it. Here is pretty much what we know:
George is not a linguist
The lack of language variation in Westeros was brought up -- lots of cultures, like the Dornish, the ironborn, the northmen -- but everyone speaks the common tongue. George admitted it wasn't very realistic, but he admitted he stole the idea from Tolkien. He admitted he could have more languages, but the books are in English and so he'd create more nonsense words to represent languages and then have to "translate" them anyway. He admitted it'd be more realistic to have seven or eight -- or even fourteen, or twenty-three -- languages, but also Tolkien was a genuine linguist, a brilliant one. He went through how many languages Tolkien made for LotR, and he says that sometimes he says when reading the books he just wants to strangle Tolkien because of how many great names he'd have, and how he could give the same thing four different names (in various languages) that all sounded great. He added that the vast majority of fantasists are in the same boat as him. Then he recounted the anecdote -- which he's recounted before -- of someone mistaking him for a Tolkien type and wanting his grammar and glossary and the like of Valyrian, and George admitting that Valyrian was seven words, and when he needs an eighth he'll make it then.
westeros, So Spake Martin, STOCKHOLM AND ARCHIPELACON REPORT
His languages aren't developed at all (see the quotes above and below)
His languages generally consist of N words
[How developed is Valyrian?]
"How little" have I developed Valyrian is the real question. I am not, alas, J.R.R. Tolkien, and I cannot imagine taking a decade to actually work up not one, but two, entire languages. I have something like eight words of Valyrian. When I need a ninth, I'll make one up.
Sorry if that disillusions any of you. It's all smoke and mirrors, kids.
westeros, So Spake Martin, WRITING DIREWOLVES, VALYRIAN, LYANNA AND HARRENHAL, AND GIANTS
If he needs another word he'll make it up (see above)
He chooses words that sound right (see first quote on Tolkien about "great sounding names")
Looking further we can see that of the known High Valyrian there are only five words, excluding nouns, with a possible sixth in the books: dracarys, valar, morghulis, dohaeris, valonqar and potentially maegi. Therefore, with 2 of these having the "val" start it is likely that this was a common start GRRM wanted to use: similar to the "ae" used commonly. As such it seems likely he just chose it cos valar sounds similar to Valyrian and sounds right.
With that said GRRM has been very open about his inspirations of which Tolkien is a massive one. For example, see this quote where he talks more about how much he's read Tolkien's works rather than his inspiration from him but this gives more of an idea of how much he's likely to be inspired from him whether consciously or otherwise:
GRRM: Well, you know there are a number of them. There is a lot of great fantasy out there. Of course, J.R.R. Tolkien, I think is the giant who looms above the entire realm of fantasy. I read Lord of the Rings when I was back in Junior High School still writing those stories for fanzines and it had a profound effect on me. I re-read it every few years, so obviously Tolkien would have to top any list of my own favorites. Another writer that I'm enormously in love with is Jack Vance, who I think is our greatest living Science Fiction and Fantasy author. Vance has written a lot more science fiction than fantasy but he exceeds at both realms. His Dying Earth, which [was] originally was published in the late 40s or early 50s, is a seminal work and he’s done some wonderful fantasy ever since. I read a Vance book the moment it comes out; anytime a new one comes I put down whatever I'm doing and go right to it. But there [is] also a lot of other good fantasy being done -- I could answer this question for a long, long time, but I think Tolkien and Vance would have to top any list of mine. They've been on top of my personal hit parade for decades now.
westeros, So Spake Martin, SECOND LIFE APPEARANCE
As such it seems likely again that while Valyrian came first valar was probably chosen because it sounds right, is similar and is the same as Valar from Tolkien's works as you suggest. It's not unknown of for GRRM to slip in references here and there, for example, the Muppets and I believe he's referenced football teams as well. So again it seems possible that the word came from Tolkien's works but no explicit mention is made.
I'm aware this probably isn't the answer you was looking for but I compiled a lot of research into this as to what we know and what answer it points to so worthwhile sharing it. I've also reached out to GRRM so if I ever get a response I'll update with a concrete answer.