Thank you everyone for contributing your ideas and for your supportive messages. I really appreciate it.
There are a lot of good thoughts in this thread. I especially liked the idea of making the trophies/medals into some kind of furniture like a coffee table and having them passed out at a kids field day to honor my uncle would be very nice too. Unfortunately, time is very limited as his apartment is in California and pretty much all the family is on the east coast. I fly back Friday so there was only so much to be done in 24-hours. He went to HS and college on the east coast and it would cost a fortune to mail all the awards to those schools or to the races he won them from. And we would need a whole extra plot at the cemetery if we wanted to bury all his trophies with him.
I called a trophy store to see if they could use them, but they said they couldn't. Salvation Army also wouldn't take them. Ultimately, the process of hauling, selling, donating, and/or throwing away all the stuff in the apartment was too much for us to take on ourselves in a limited amount of time. We hired a company that specializes in this type of thing. They come in and haul all your stuff away for you and do their best to donate everything they can and what isn't donated gets recycled.
Supposedly, 2/3 of what they take gets donated. In our instance, they were impressed by how much stuff we had that could be used by someone else (there was a lot of great stuff that's just not practical to bring on a plane with us) so I think maybe even more than 2/3 of it will find a new home.
Specifically concerning the trophies/awards, I took a few and so did another family member. I also took some pictures of them on the shelves (although I discovered as we cleaned out closets he had boxes and boxes of trophies/medals not on display). The rest will go with the hauling company.
While this isn't the optimal way to honor my uncle, the company did say that sometimes organizations will come in and take old trophies and repurpose them for their own events by removing the plaque and adding their own. And even if that doesn't happen, the idea that the trophies could possibly be "recycled" is some solace for me. I don't know the science behind recycling, but if the materials from his awards can be melted/broken down and used to build new things, at least they'll have served a purpose, rather than being buried in a landfill. Either way, his accomplishments will live on in the record books, with or without a trophy.
TL:DR: Thank you everyone for your advice and support. We kept some for ourselves and gave the rest to a company that tries to donate your old stuff and recycles what they can't donate.