I have just broken the Yoshi Valley NSC Flap world record with a time of 0'30"02. I have a photo for proof but no video of the run. Is it possible for me to still submit the run?

The place to submit time trials is mariokart64.com Unfortunately you're looking at the wrong time though. You're playing on an NTSC system while comparing to the PAL world record. The current NTSC record is 26.01. The current PAL WR of 30.92 is equivalent to a 25.72 on NTSC.