In the 1990 film, the Turtles order their pizza for delivery to "122 ⅛ᵗʰ". This works to their advantage since the famous Domino's 30-minute-guarantee (and relative difficulty in finding the sewer address) means that it ends up being free. They do still tip the guy though.
In the second film, they order their pizza for delivery to April's apartment. Since she's out at the time, the plan seems to be to break into her apartment (via the fire escape) and then accept the order at her door, presumably wearing some kind of rudimentary disguise.
The pizzeria owner notes that she's becoming one of their best customers lately (referring to her by name) so it seems that under normal circumstances she's the one to accept the orders. In this case, their plan fails and they simply take the pizza from the pizza-boy after he leaves it lying around.
In the 2007 CGI remake, the original script contained these lines;
MICHAELANGELO : KIDS BYE!!!!!!
ON DOOR - As the door closes and Mom turns back, a green hand reaches
in quickly a grabs a box of pizza. The door closes.
MICHAELANGELO (O.S.) : Heheh... score.
This scene was cut from the film. There are plenty of pizza boxes but no explanation of how they got there.
In the IDW comic series, they seem to rely on April or Casey Jones doing the collecting;
and in the Gameboy game, the pizza slices simply appear in mid-air