And no, it is possible. Here is an example that does it with BufReader, which has T: ?Sized and uses it as a field: https://play.rust-lang.org/?version=stable&mode=debug&editio...
Though it comes with a caveat that you can't take self by value, which is perfectly fine for this use case & is what a normal allocator-aware language does anyway.
That said, other than the problem of this being a breaking API change for Read::read_to_end, another problem is that Vec's layout is { RawVec, len } and the allocator is inside RawVec, so the allocator is not the last field of Vec, which is required for structs to contain unsized fields. It would require reordering Vec's fields to { len, RawVec } which may not be something libstd wants to do (since it'll make data ptr access have an offset from Vec ptr), or to inline RawVec into Vec as { ptr, cap, len, allocator }.