This is definitely something you can DIY.
Two things to keep in mind.
First, you get what you pay for. Go to a real tile store and get a good quality sealer.
Second, make sure you get the correct sealer for your tile. There are different sealers for different tile - ceramic, natural stone, granite, etc.
Unless this is a stone floor, you only put the sealer on the grout joints, not the tile. Anybody can do this, it's like painting lines with water.
You can use a 3" foam brush if you want and apply 2 coats. Wrap the brush in aluminum foil and it won't dry out between coats.
"They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety."
B. Franklin 1759