Though Dota Underlords doesn't have a way via the U.I. to check your match history, it turns out there is a workaround.
/u/Brassica_Rex came up with the following.
Just go to your steam profile>games>dota underlords>personal game data>games>matches
It's not great, but it's the best we currently have! Here is a shot of what it looks like if you want to give it a go. Quite a neat, quick little hack if you ask me.
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