Even modest riding by a 4-year-old can be enough to crack and tear away the pegs from the seat of a Big Wheels tricycle. The manufacturer's replacement is $13 -- and just as likely to fail again! So instead, for around $4-$5 (excluding cost of glue) I built a replacement out of PVC pipe.
Step 1: Parts and Assembly
The measurements below should be taken as rough guides. Note that there are slight variations among manufacturers of PVC pipe and fittings, some may be more "loose" or "snug" than others, and/or fit more shallow/deep into the fittings, so you'll want to "dry fit" your pieces before gluing.
(1) 2' length of 1" Schedule 40 PVC
(2) 1" PVC Tees
(2) 1" PVC Ells
(2) 4-1/2" horizontals
(2) 1-3/4" verticals
(2) 2-3/4" pegs
Optional: A simple but very useful "jig":
a piece of scrap lumber cut to 4-3/4" width
Dry fit the lower portion (2 pegs, 2 tees, 1 horizontal) and confirm that the resulting dimension between the inner edges of the pegs is 4-3/4" (or that the center-to-center dimension is 6", a little harder to measure, but more accurate). Insert the piece on the Big Wheel to confirm proper fit. Mark the exposed pipe along the edge of fittings to serve as a stop-guide during glue-up, and confirm that your jig (if using one) produces proper spacing.
During glue-up, use your stop-guide marks and jig to prevent inserting the pipe too far into the fittings (the pipe often slides more easily and deeper into fitting once glue is applied). Work quickly, PVC glue (cement) sets up almost instantly. Have a flat surface upon which you can lay the piece during assembly to prevent "racking" (that is, to keep the Tees and pipe "flat").
Assemble the lower portion, then assemble the upper portion, verify that they line up properly, then assemble top to bottom with the vertical pipe sections.