If my immigrant wife is already in the U.S. with me, but not a citizen yet, will me filing for unemployment cause any problems?

No it won't but you will still need to meet the minimum income requirements based on your household size or if you can't meet them then qualify on the strength of your assets. If you can't meet either of those tests, then you will need to enlist a joint financial sponsor. Additionally there's a new declaration of self sufficiency form that the beneficiary must fill out that tests whether they are likely to become a public charge. Work with an attorney to help you so that your case is processed correctly. Counsel anywhere in the USA can represent you and some of us charge a very affordable flat fee.