Both are acceptable. Grammatically, they change things a bit but they mean the same thing. The second one also sounds a bit strange because you're using "to" twice so I think that's why we drop it in natural speech. Also, the second one may also be stressing why you want to go since it's used as an adverb modifying "to go" (see below for that explanation).
1) I want to go look for my wedding dress today.
- Subject: I
- Verb: want
- Direct Object: to go look (infinitive noun form of compound verb "go look")
- Indirect Object (as a prepositional phrase): for my wedding dress
- Adverb: today (Note that this adverb is modifying the infinitive and not the main verb.)
2) I want to go to look for my wedding dress today.
- Same as above except now "to look" is an adverb infinitive modifying the noun infinitive "to go" (which is still the direct object like above)