The "a" command expects to be followed by a "", so it's eating the one in your first "\t". If you add another "" it seems to work as you want it to:
$ echo string | sed '/string/a \tstuff\t\t\tmorestuff' string stuff morestuff $
Ah ffs, lol... It would also help if I emailed what I was typing. I use so many versions, I get mixed up. I typed the email with <'> but wrote <"> in the console which is why I couldn't make it work.
'/string/a \tstuff\t\t\tmorestuff' != "/string/a \tstuff\t\t\tmorestuff"
Thanks Filipe! jlc