Yes you can put it in a garden sprayer. I’ve done it lots of times. Thomson water sealer comes in a water base. But you can also use a “chemical” based sealer, stain etc. There are water based stains to be had.
For small jobs and or ones I don’t mind refilling when needed. 1 gallon sprayer can be had at wall mart for $5+. Larger ones for not much more. The spray nozzle can be adjusted/opened enough for thicker stuff.
But the cheap sprayers have plastic spray wamd and plastic nozzles. So, depending on the chemical composition of what is being sprayed there is a possibility the plastic components, esp the nozzle may start to “melt”, gum up, break down. It also depends on how much is being run through it. A real cheapo sprayer may only serve once, but should last long enough to get the job done. They are meant to have garden chemicals used in them. You can get inexpensive ones that have a metal wand and spray nozzles. Some come with several different "tips"nozzles to suit the job.
The cheap ones I have bought and used work fine for the cost. The internal hand pump seals give out before the rest does. I bought a decent size one with metal wand and nozzle. I think it was around $30. The pump seals gave out. So I drilled a hole in the top/side and put a tire valve in. I have a small air tank I fill with my compressor. Take that with me to pressurize the sprayer. The sprayer has a pressure release valve if I fill it too much. Most of the better ones do.
I fill garden sprayers with disinfectant for foaling stalls, stalls, barn walls etc. Soap up barn walls and then spray off with a hose. Works great if one doesn’t have a pressure washer.