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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Make an InfoPass appointment at your local field office and inquire about receiving a temporary green card stamp.
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.