How can I use Apple Pay if my side button does not work?

There is very little you can do if the power button does not work. You cannot even force restart it. I presume you have raise to wake in order to open the phone but you can't even turn it on if you ever had to shut it down.

You should arrange a repair 

iPhone Service & Repair - Apple Support

Or

 Contact - Official Apple Support

Anything other than Screen or Battery may be a replacement phone and the price is in the article under "Other damage"

See also:  Get your iPhone, iPad, or iPod touch ready for service - Apple Support

It needs repairing, which if Apple does it out of warranty will be a discounted price t phone.