Can I get an additional remote key for my 1999 Toyota Camry

Yes, they sell them at parts stores, dealerships, and online.

Like @brotherben said, you definitely can. Copy and paste the following address to visit a site online. They offer free instructions for pairing and fob model numbers. (I use the site for info but generally buy from Amazon)...https://northcoastkeyless.com/1999-toyota-camry-solara- keyless-entry-remote-programming-instructions/

Yes, but I highly suggest you avoid the cheaper route of buying a key fob via Ebay, especially those shipped from China. I made that mistake with an old car, I purchased two, but neither would access the LEARN mode. I paid $30 for two, but should've spent the $50 for a single good brand. If you purchase one from a reputible store, they should be able to program it for you. A dealership can do it as well.

I trust Northcoast Keyless. The important part is making sure you get a fob with the right FCC & Part Number (FCC: GQ43VT14T, P/N: 89742-06010) but not everyone lists the fob with these part numbers, just the vehicle. I would recommend programming yourself to save the money.