Afghan Allies Protection Act of 2023 This bill makes an additional 20,000 special immigrant visas available to eligible citizens or nationals of Afghanistan who assisted U.S. efforts there and addresses related issues. (Generally, these visas are available to individuals who were employed in Afghanistan by the U.S. government or the International Security Assistance Force for at least one year.) Specifically, the bill expands the window of eligibility for meeting certain requirements for such visas. Currently, an applicant must complete the employment requirements by the end of 2024, and the bill extends this window to the end of 2029. Similarly, the bill extends the deadline to apply for a visa with the Chief of Mission from the end of 2024 to the end of 2029. The bill also makes these visas available to otherwise eligible individuals who could not complete the employment requirements due to an injury sustained during the employment. The bill also addresses the processing of these visas, including by (1) requiring each relevant federal agency to designate a senior official to oversee the integrity and efficiency of visa processing; (2) authorizing the use of virtual video meetings to sign applications and to give oaths; (3) requiring the Department of State to reimburse applicants for the cost of medical examinations required by immigration law, if the applicant petitions for such reimbursement; and (4) requiring the State Department to develop a strategy to address pending applications.