How can I make a payment to a vendor that only accepts bank transfers from my credit card, paypal, money bookers, or a similar service? I'd like to do this completely online, without going to the bank.
You'll have to call your credit card issuer and ask them. Generally, credit cards don't do bank transfers, since its not a bank account per se. But it may be in some cases that there's an underlying bank account over which your credit card is managed, and then they might be able to do something like that.
But we won't know, only your card issuer will.
I would strongly advise that you do not use Western Union or any other wire transfer service for any money transfer, unless you personally know and trust the recipient of your funds, and the method by which you can communicate the details of the transaction. As was mentioned, wire transfers are irreversible and very difficult to trace to ensure the actual recipient got the funds. In some countries and situations, literally anyone can show up at any Western Union location, correctly recite the details of the transaction (which is proof enough they are the intended recipient) and receive the cash.
If the vendor accepts PayPal, then simply set up a PayPal account, linked to your credit card, and pay the vendor that way. This can be done 100% online, assuming you have a valid card.
The only way I can think of to do this would be to take a cash advance against the card, then buy the wire transfer with that cash. Of course cash advances typically start accumulating interest immediately, since the credit card company isn't capturing a fee from the vendor, and may be at a different interest rate than other charges... so given the choice, I'd suggest you write a check instead.
As @KiethS points out, there have been a number of scams involving getting people to send (non-cancellable) wire transfers and then not delivering the goods/services paid for. A wire transfer is, essentially, cash. If you don't know exactly who you're dealing with, don't.