It's "kind of" possible through using gdb and setenv(3), although I have a hard time recommending actually doing this. (Additionally, i.e. the most recent ubuntu won't actually let you do this without telling the kernel to be more permissive about ptrace, and the same may go for other distros as well).
$ cat setfoo
#! /bin/bash
gdb /proc/${PPID}/exe ${PPID} </dev/null
call setenv("foo", "bar", 0)
END
$ echo $foo
$ ./setfoo
$ echo $foo
bar
