Is there a financial institution that loans less than $5000 for motorcycles that most veterans use?

PFC James (LURCH) Janota
At that price Im assuming it is a older used bike?
If so I think you will find few if any banks or credit unions will do a secured loan for it.
Wont hurt to ask the usual suspects USAA Navy Federal and the bank if different you bank with now.
And of course a non secured loan will come with higher interest rates..something to avoid for what is likely not a Need, but a Want.

I have on occasion used a CC to make a large purchase when I knew the cash would be available in just a few weeks or less....but its not something Id do if i needed to make payments on it for more then one billing cycle.

SGM Erik Marquez beat me to it, take a look at your credit union, you may need to take an unsecured loan

Harley Credit will finance a used Harley. Not sure what you're looking at.