Ugh, Assuming you have the transplants grown and are ready to transplant them?
plantel trays are designed for a semi automated transplant system.
Some systems exist where you have one or two people
feed transplants into a machine pulled behind a
tractor driven by a third person.
Generally though the machines are designed for a specific size of plug. Like grown in the plantel trays.
Those machines are also spendy. May be of limited availability this year.
Easiest would be to direct seed the seeds. Then you just need a seeder. That's something I do to get lots of tomato plants. If you have to start over because of plug size issues. And you have a long
enough season to be out planting now, that's an option though it will hurt to loose that many starts.
Don't blow out your back though.
Hand planting tool thoughts. Keep your back straight. Stand up as much as possible. Avoid hand trowels. Use long handled shovels and long handled eye hoes to dig. Dig a row of holes. Drop a row of seedlings, then backfill with long handled shovel. Remember with tomatoes you can bury some stem. If your back gets bad stop and rest, stretch. If need be
compost some seedlings and direct seed the rest. Your back is more important.
With just a furrower you might be able to dig the whole trench, drop the seedlings in it, then backfill.