Can you use thompsons water seal with a Wagner model 220 power sprayer?

Yes,...but why you would use that product is the Question.
There are many superior products that will last longer. You need to stop the wood oils from drying up to protect your wood products.
Thompsons is very temporary and hard to apply evenly. You will get 1 to 2 years from it before recoating is needed.
Oil stains or oil based wood treatments wood be best for long term look and durability. It is best to find a product that will block UV rays as well as have a Mold resistant quality to it.
There are also some very good Water based products available for outdoor use.
I believe Oil is the way to go though.