Short answer: yes, He can forgive if we make a vow and fall short. I'd specify that this is from [insert denomination] perspective, but I think this is agreed upon by all mainstream denominations.
Longer answer, with support:
Yes. Even to those that believe you can "lose your salvation", you can repent and confess it. To those that don't believe you can lose your salvation, the "yes" is even more obvious. We believe that He has forgiven all our sins, past, present, and future.
The central message of Christianity is that the reason we need savior in the first place is that we fall short and cannot save ourselves. We can't possibly be "good enough".
Finally, 1 John teaches that we all sin. If we try to claim otherwise, we're liars, and that instead if trying to claim we have no sin, we should just confess it, and know that He is faithful and just to forgive us our sins.
1 John (KJV)
5 This then is the message which we have heard of him, and declare
unto you, that God is light, and in him is no darkness at all. 6 If we
say that we have fellowship with him, and walk in darkness, we lie,
and do not the truth: 7 But if we walk in the light, as he is in the
light, we have fellowship one with another, and the blood of Jesus
Christ his Son cleanseth us from all sin. 8 If we say that we have no
sin, we deceive ourselves, and the truth is not in us. 9 If we confess
our sins, he is faithful and just to forgive us our sins, and to
cleanse us from all unrighteousness. 10If we say that we have not
sinned, we make him a liar, and his word is not in us.
However, I'm with you on refusing to make vows to God. Scripture says it's better not to as well.
Ecclesiastes 5:4-5 (KJV)
4 When thou vowest a vow unto God, defer not to pay it; for he hath no
pleasure in fools: pay that which thou hast vowed. 5 Better is it that
thou shouldest not vow, than that thou shouldest vow and not pay.
And, in general, we are to keep our word, so of it is in our power to fulfill our vow, we should. There in nothing in Scripture that relieves us of our responsibility to pay what we owe, if it is within our ability to do so. I'd question whether "repentance" is genuine if the person in question failed to do whatever it takes to try to make it right.