jemanden is the accusative case, jemand is nominative. Jemand is the grammatical object to finden, and the accusative is the correct choice here, although I can imagine jemand might be acceptable in some dialect.
Both finden and suchen are possible, but note that they do not quite mean the same - "finding" is successful "searching".
You do not need an extra "um". In fact, if you were to include it, the sentence would end up sounding strange:
Um mein Auto Auto zu waschen, kann mir dieser Jemand helfen. (In order to wash my car, that "somebody" can help me.)
The relative sentence um mein... would likely be interpreted to refer to what precedes it directly, ie to jemanden. If you change the word order, the sentence sounds alright to me even with the um.
Um mein Auto zu waschen, will ich jemanden suchen/finden, der mir helfen kann.