Paying additional license fee for library music just for our own websites?

[JEK3]

I think the answer lies somewhere in the contracts. Try to read them again and look if you can find the rules regarding demo purposes. I'm sorry I can't give you a better advice but it really depends on the kind of license you bought. For instance, if the contract is giving you the right to use the music only in Vimeo, then the answer to your question would be no. If the contract gives you the possibility to use the music for "the web" without specifications, then the answer would be yes.
Depending on the library, sometimes their contract is online and can be viewed without registering. If that's the case you could link it here so it'll be easier for GS to help you.

[drBill]

I'd call the library directly and ask. My guess is that if you give a credit to them, and are using it to demo and promote yourself, they would be more than accommodating to let you use it in hopes that you will purchase from them again.

[MusicTrax]

There may also be a limit on segment times. I seem to recall some library music had buy-out rates, feature rates, home video rates, and web rates, plus various schemes that would cover all of the above. And then if you just used :20 or :30 seconds for a promo, the rates were very, very inexpensive, like $50-$100. But it depends on the specific company.

I think pretty much all of them want each usage to be reported, whether or not there's a fee involved, just so they can keep track of it.

[Jeff Hayat]

As said, it varies from lib to lib, and client to client.

Some libs for some of their cat offer what's called all media worldwide in perpetuity. Meaning you can use those tracks anywhere, for any purpose (TV, DVD, theater, web etc) and with no time limit. Some cats have certain restrictions, depending on the license you purchase.

Being that it varies so much, the only way to get a real, proper answer is to follow Bill's advice, and get in touch with the library directly and ask them.

Cheers.