No, you would not file a DE W-4 with PA. You will get credit on your DE return for taxes paid on the same income in PA, but it's often not a dollar for dollar credit.
Essentially, PA has a convenience of the employer rule. If a nonresident earns income from PA sources and the income is earned for the convenience of the employer, then the PA income would only be taxed in Delaware. But if the wages from PA sources is for the convenience of the employee, then the employee needs to file a nonresident PA tax return for those wages and claim a credit on the DE return for those taxes paid in PA on their resident DE return. Here is an FAQ link on filing a nonresident return for an out-of-state employer.
Since you don't want to have DE tax withheld from your wife's paycheck, the simplest option is to make DE estimated tax payments instead. You would put the dates and amounts of payments on your DE return and any excess would be refunded.
