I think you can buy it online, see below.
•You cannot make a purchase online or by phone if:
◦You are a resident without a valid New York State driver's license.
◦You are applying for a reduced fee license as a veteran with service-related disability of 40% or more.
◦You are applying for a free license as an active duty military member under the Patriot Plan.
◦You are a first-time hunter or trapper applying for a hunting or trapping license.
•Customers under 16 years old must have a parent or legal guardian assist with online and phone purchases, and validate all information at checkout.
•Allow for 14 business days to receive your license or DMP tags by mail.
•You will receive a receipt of purchase immediately after your transaction:
◦For hunting and trapping: this receipt is NOT valid proof to perform any hunting and trapping activities. You must wait to receive your license in the mail before going hunting or trapping.
◦For freshwater and saltwater fishing: the receipt can be printed and used as valid proof to perform fishing activities until you receive your license in the mail.
According to the above, you can buy just a fishing license as long as you have a valid driver's license.
When you click into the decals sections it says:
"If you are a new customer, click here to register as a DECALS customer" and then you register.