What age??? I know Barrett-Fletcher use to have the 14-17 year old championship game played in the dome in T.B./St. Pete. They also play most of their games at the minor league or training camp facilities. I have not looked at their tournament down in Tampa Bay for a few years so things might have changed.
Some of the showcase tournaments do for the older boys. I believe Guerry took the 16U Astros up to one and played in Wrigley Field. Also in Minneapolis I believe. Mostly those are Perfect Game or BCS Regionals, I think, though.
Keep in mind, the times you would want to play coincide with when the teams are using them. Spring training facilities are more common as they do not use them as much in the summer.
Back in the late 80's and early 90's we used to play tournaments in alot of minor league stadiums, Memphis Chicks field was astro turf infield, played in Engle Stadium the old Chattanooga Lookouts stadium, ceter field was 471. Played in Montgomery Alabama, Patterson Field it was home to the Detroit Tigers AA team, NCAA division 2, world series. We even played in the Atlanta Federal Penitentiary against inmates.