Lawful permanent resident of United States like yourself does not need to apply for a permit to travel internationally. What you instead need to do is schedule and infopass appointment at your local district office, and inquire as to why the processing of your I-90 taking so long AND also obtain a sticky in your current green card or stamp in your passport, extending your green card status for a few months, enabling you to travel and be readmitted to the US. Make sure to bring your airline ticket reservation or other proof of a date certain travel with you to that info pass appointment.
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You can schedule an Infopass appointment to receive temporary proof of your LPR status while you are waiting for your replacement card to arrive in the mail. Take a copy of the I-190 receipt notice and your unexpired passport to the appointment.
Have you had any trouble with the law since becoming an LPR?
Yes, you may apply if needed. But, I would caution you to travel out of the country with the expired green card.
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I agree with my colleagues. Try an Infopass appointment.
Disclaimer: This answer is for informational purposes and does not take the place of a consultation with an experienced immigration attorney. This answer does not create an attorney-client relationship.