Yes, Spike Feeder's ability can target itself. There's no language in its ability preventing it from doing so. When a permanent's ability cannot target itself it's typically worded as "another target..." like you see in cards like Avacyn, Guardian Angel.
May 11, 2018 10:41 p.m.
Although, I should point out, that if removing the +1/+1 counter from Spike Feeder causes its toughness to drop to zero it will be moved to the graveyard as a state-based effect before the ability can resolve. Removing the counter is part of the payment to put the ability on the stack, and SBAs are checked every time a player gets priority.