Emergency Stopgap USCIS Stabilization Act This bill expands Department of Homeland Security (DHS) authority to provide premium processing services for certain immigration-related applications and contains other related provisions. DHS may collect a fee to provide premium processing of an application for any immigration benefit that DHS considers appropriate, subject to certain requirements. Currently, DHS only has statutory authority to provide premium processing for employment-based applications. Under current law, DHS must use such premium fees to provide premium processing services and make infrastructure improvements. Under this bill, such fees shall be used for these purposes and other activities that offset the cost of providing adjudication and naturalization services. DHS may suspend premium processing of applications only if circumstances prevent the timely processing of a significant number of such applications. DHS shall provide those who have requested premium processing with access to case status information and communications channels to the premium processing units. DHS may expand premium processing to certain immigrant benefits and set fees for such processing without following certain rulemaking procedures if DHS meets certain requirements, such as limiting the premium fee to specified amounts. The bill also increases the premium fees charged to applicants. DHS may, subject to requirements, biennially adjust premium fees to reflect inflation without following certain rulemaking requirements.