I am not an active TTM'er, but kind of follow the scene a little. I keep saying I will do more, but have only sent off maybe a couple dozen requests in 20 some odd years. Frankly, I'd rather buy them then go through all the work and the waiting. Maybe some day...
There are a few sources of TTM autograph addresses that seem to be most popular. These are all fee based (join the site, buy they list, etc), but seem to stay the most current with who does and doesn't sign, how much they charge, etc. A lot of players still will sign for free or there are recommended donation amounts. Of course, anyone can look up addresses on their own or send c/o teams and try to send for no cost, but the odds drop considerably. Out of the few I have done, several were obscure players I tracked down online and reached out to directly.
Jack Smalling has an address book
http://www.baseballaddresses.com
then there is Harvey (can't remember the last name)
https://sportsaddresslists.com
and then SCN
http://www.sportscollectors.net
Pastpros is a newer site that seems to be growing with players:
https://pastpros.com
A kid sending letters will probably have better luck than an adult writing or typing a letter, but some of these guys get so much mail that it may not matter either way. I do know that Ryan does have a mail in service that is relatively cheap considering his contributions to the game and his superstar status. Want to say around $25 per signature.