Yes, you have to wait to request a refund; there’s no exemption available from withholding.
According to the city’s Wage Tax (employers) page,
All employers located in Pennsylvania are required to register with the City of Philadelphia within 30 days of becoming an employer of either:
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A resident of Philadelphia, or
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A non-resident of Philadelphia who performs services for an employer in Philadelphia.
Employers are required to withhold the Wage Tax from all employees who meet these criteria.
Your only recourse is to request a Wage Tax refund. Beginning in November 2021, Wage Tax refund requests have to be submitted through the Philadelphia Tax Center.
Please see Wage Tax refunds for more information.