You didn't say which race weekend you are attending, so I will answer as if you are going to Star Wars. Kids races are on Friday and Saturday. Here is the Expo schedule:
runDisney Health & Fitness Expo
ESPN Wide World of Sports Complex
10:00 a.m. – 7:00 p.m. on Thursday, April 14, 2016
9:00 a.m. – 7:00 p.m. on Friday, April 15, 2016
10:30 a.m. – 4:00 p.m. on Saturday, April 16
If you get to the Expo BEFORE it opens on Friday, i.e. around 8:30, you will probably be in line early enough to get their stuff. In February, the line went all the way out to the parking lot and the wait was around an hour or more. For me, it would be cutting it too close for comfort but if that was the only way I could do it due to my arrival day then I would make it work by arriving VERY early. For a 9 am opening, people will begin lining up around 8. Once you pick up their packets, you may have to go to the Expo building to get their t-shirts. I can't remember if kids shirts were in the packet pick up bldg or not. On the other hand, if you are doing kids races on Saturday, then no, it will be next to impossible to get done in time if you go on the day of.
Lynda