What do you want to eat today?
In other words, what would happen if I say:
(I want to know the rough meaning of the sentence even if the sentence is wrong.)
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You can say...
を is the object marker. You can use が here instead, since が can mark the object of the ～たい form verb.
Your second sentence:
would be incorrect. You don't use the topic/contrastive は for a question word.
◎「何が～～？」 ◎「誰が～～？」 ◎「いつ(が)～～？」etc.
×「何は～～？」 ×「誰は～～？」 ×「いつは～～？」 etc.
As you know, は is basically the topical/thematic particle (主題の「は」) or the contrastive particle (対比の「は」), and can replace が, を. But it can't be attached to the question word 何、どこ、誰、いつ etc... (cos 何、どこ etc. can't be a topic/theme of a sentence, nor can they be contrasted with some other element of the sentence.)
While both 「は」 and 「が」 as particles are "subject indicators" (please take note that I am not sure of the actual linguistic term for this), they emphasise different things.
「は」places emphasis on what comes after it, ergo:
(I feel) good now.
(I) am tired today.
In your hypothetical example, you'd be placing emphasis on 「食べたい」, as in "What do you want to eat?" It feels like asking if you want to eat "what".
As ericfromabeno states (along with Chocolate's very helpful answer), using 「は」for this kind of statement is unnatural. I think 「を」is still more appropriate particle to use in this sentence.
「が」places emphasis on what comes before it, ergo:
"I like chocolate."
"I feel bad."
Your hypothetical example places the emphasis on 「何」, as in "What do you want to eat?"