First of all, an inquiry isn't a big deal - and any lender you work with will require that they pull your credit; it will hit your credit for about 3-5 points but it needs to be done (inquiries from multiple mortgage companies will only effect your credit score as if there were one inquiry as long as it is within a 45 day period - 30 days just to be safe).
Secondly - do not get a new credit card. New credit negatively impacts your score because there isn't an established history for that account AND you'll have another inquiry. A new account will have a negative impact to your credit score for about the first 6 months and a neutral impact for months 7-12 after which the history of the account - assuming it's good - will positively impact your score.
VA guidelines do not have credit score requirements but there are requirements regarding credit history. The investors who buy the loans have guideline overlays that often implement score requirements. 640 is quite common. It would probably be good to work on getting the score up before you secure a loan since you may otherwise have to pay for a loan-level pricing adjustment due to your lower score. I have a credit repair company That I work with who has done a good job for many of my clients. Feel free to contact me if you would like me to put you in touch with them.