I like to run website with out pc running 24x7
There are some options to do this as Bushman mentioned.
Is this possible to have Davis connect to router and upload to site ?
Yes using WeatherlinkIP or, as Bushman said, the new WiFiLogger alternative. This logger uploads to WU, PWS, CWOP, Weatherlink.com, AWEKAS. Or you can also use it with 3rd party software like Cumulus and then to your own website but this would need a computer running.
What web hosting recommendation ?
I use GoDaddy and reasonably happy with them.
Enjoy,
Paul