I want to fully tile an 8x8 board using the 19 fixed Tetrominoes, allowing repeats of Tetrominoes. For example, this is a valid solution:
Here are the 19 fixed Tetrominoes:
I am looking for a list of all solutions or an algorithm that can find all the solutions. Or a link to one of the known solvers
Since you allow repeats of the tetrominoes it does not really matter whether you consider them fixed or not. You are simply filling the 8x8 square with tetrominoes, regardless of their shape and orientation.
I think there are probably several billion solutions. If you fill exactly half the board, a 4x8 rectangle, then there are $40899$ solutions. Putting any two such solutions together as an 8x8 square we already get $40899^2=1,672,728,201$ solutions for the square. There will be many more solutions without a fault line along the centre. Of course, if you reduce by symmetry, i.e. considering solutions that differ by rotation/reflection the same, then there will be about one eighth as many solutions (slightly more due to solutions with symmetry).
There are many polyomino solvers out there to explore these problems. I use my own, which you can download here: Polyform Puzzle Solver. It is not the fastest, but quite flexible.