is it possible to get a state court appointed attorney to represent me in my patent infringement case

No. Our federal and State governments only provide legal counsel to indigent persons accused of criminal activity. Patent infringement is a civil wrong, not a criminal violation.

Patents are litigated exclusively under federal, not state, jurisdiction. You'd have to take your patent infringement case to your state's federal district court; state courts cannot hear patent infringement matters.

If you have an infringed patent, you really don't want an appointed attorney litigating the matter for you anyway. Patent infringement is a real concentration for litigators, and you want someone who has experience in litigating in this arena. You wouldn't hand a dermatologist a skull saw and tell him to do brain surgery, would you? This area of the law involves a similar level of concentration.

Your question makes me suspect that you're perhaps somewhat short on funds; you might see if you can find an intellectual property litigator who will take your case on contingency (you win, they get paid handsomely out of the settlement; you lose, you still pay for expenses incurred in the case but no attorneys' fees). There are people out there who handle patent litigation on a contingency fee basis; try a GOOGLE search on "patent litigation contingency" and see what comes up.

Good luck.

NO. Patents are exclusively federal jurisdiction. States have no say.
DISCLAIMER: This answer does not constitute legal advice, and should not be relied on. Each question is fact specific and requires a comprehensive evaluation and consideration of all the facts and documents at issue. This answer does not create an attorney-client relationship.

Typically state lawyers are only appointed in criminal cases. Legal aid lawyers help low income people with civil matters. Patent infringment matters typically would not qualify for these type of assistance programs. However with the current economic state many lawyers are offering alternative billing arrangements including contigency, partial contingency, and flat fee. You may be able to benefit from one of those types of billing arrangments. Additionally, you may considering partnering with someone such as one of the infringers competitors to fund the litigation. There are issues with all of these solutions so you should consult with a lawyer you trust before entering into any of them.