Can they refuse partial payment?

Yes. They will want a one-time cash settlement of the claim, perhaps with a discount; or a monthly payment arrangement, if you are lucky. You could verify the name and address of the collector and the creditor, and send a partial payment to see what happens. By doing that, the collection agency will have the name of your bank and will facilitate a garnishment of your account(s) if it gets a judgement later on. So, I would send a money order, and keep the receipt and cover letter in a safe place.

Technically they can refuse any payment, what ever is in their terms and conditions. But I would assume they would love some money, and would be happy that you made a partial payment

They can refuse a partial payment UNLESS its either a minimum payment according to the terms of the contract or part of an agreed upon settlement. Frequently they will refuse partial payments in order to pursue legal rememdies, like garnishments.