It is wrong. Modal verbs, like can, might, would etc. are "defective", they have no participle (canning, mighting) and no infinitive ("to can", "to might")
A modal verb "would" is followed by the bare infinitive for example "be":
He would be happy, if he saw his family more often.
But might has no infinitive, so "it would might use..." is incorrect. You can search for "double modal" for further details on this.
You can use other ways of indicating the "might" modality. In this case the adverbs "possibly" or "perhaps" has a similar meaning.
I thought it would possibly use an API.
or just using "might" on its own is correct, with the same meaning as "would possibly"
I thought it might use an API.