That seems to be (an expansion of) a little bit from a poem, Dumt fôlk by Gustaf Fröding: https://sv.wikisource.org/wiki/Dumt_f%C3%B4lk.

But that only has "...e å, å i åa ä e ö" -- not the bit about "å i öa ä e å" from that tongue-twisters page, which seems to be a later addition. Ungrammatical, too: I'm sure even värmlänningar would say "på öa", not "i öa".

(Oh, and "e" is of course a contraction of the indeterminate article "en", a/an.)

Further: e: dialectal version of "det", i.e. "it" öa, åa: adding an a on the end of a noun is a dialectal way of expressing "the 'ö'" and "the 'å'", respectively.

But the 'e' doesn't precede any of the forms 'åa' or 'öa', but only the indeterminate 'å' and 'ö'. The 'e' is definitely 'en', not 'det'.

Wouldn't "e" just be a contraction of "en" in this sentence? Is it "det" in the dialect?

No, you're right, it's "en": "...en å, och i ån är en ö". "Det" would be totally ungrammatical, dialect or no dialect.