Gol><>, 20 bytes

iE;:`#=Q$`~P%o|`;=?P

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How it works

iE;:`#=Q$`~P%o|`;=?P

                      Accumulator (m) is always stored at the bottom
iE;                   Take input (n) as char, halt if EOF
   :`#=Q      |       If n == '#'... (preserving n on the top)
        $`~P%o        Swap m to the top, output m%127 as char
               `;=?   If n == ';'... (discarding n; m is exposed to the top)
                   P  Increment m
                      Repeat indefinitely

Gol><>, 20 18 bytes

Credit to Jo King.

iE;:`#=q$o`;=+`~P%

Try it online!

How it works

iE;:`#=q$o`;=+`~P%
                    Accumulator (m) is stored at the bottom
iE;                 Take input (n) as char; halt if EOF
   :`#=q            If n == '#'... (preserving n on the top)
        $o            Output m as char, discarding it
          `;=+      If n == ';', increment m
              `~P%  m = m % 127
                    Repeat indefinitely

Previous submission, 20 bytes

iE;:`#=Q$`~P%o|`;=?P

Try it online!

How it works

iE;:`#=Q$`~P%o|`;=?P

                      Accumulator (m) is always stored at the bottom
iE;                   Take input (n) as char, halt if EOF
   :`#=Q      |       If n == '#'... (preserving n on the top)
        $`~P%o        Swap m to the top, output m%127 as char
               `;=?   If n == ';'... (discarding n; m is exposed to the top)
                   P  Increment m
                      Repeat indefinitely