I'm going to start by saying that this is, for me at least, a highly confusing topic. The whole business, as a whole, of embedding fonts into digital applications is a relatively new process and the models of distribution are very much convoluted... You have a plethora of different usage licenses, one of which seems to be offered in a completely different way to the rest. I'm of course talking about embedding fonts into websites. The legality of what you can and can't do with fonts is different everywhere... it's quite annoying and I can't wait for the first company to offer a much simpler font service that caters for all usage scenarios....
Anyway, on to the solutions. You have three options:
1. Explain to your client that in order to do this you will need to pay a fee for using a premium, licensed font and then go to one of the font services that will charge you a monthly fee. (This is really for people who have a lot of clients and use fonts on a regular basis)
2. Embed the font into the website using CSS, without permission (not recommended, especially for commercial use)
3. Choose another font.
If you don't want to set up an account with a font provider for a monthly fee then I recommend that you choose another font. It's not really worth your while to pay a monthly fee for a service if you're only using one font. I don't know what your client is like, but some can be quite particular and unwilling to make changes, so you will need to convince them in some way. Luckily for you, the foundry that designed Calibri released a free font called Droid Sans which looks very, very similar to Calibri, so much so that I doubt your client will notice. So, if you believe that your client will pay the extra money for the font, but you don't want to commit to a monthly payment for the fonts then you could just switch the fonts without telling the client.
Obviously you know your client better than I do so choose what's going to work for you, and not bite you in the arse later.