These situations can get complicated. You will need a valid and unexpired passport from you nation of citizenship. You will also need to file and secure the proper paper work from USCIS. It may be worth pulling out a credit card and seeking assistance from an experienced immigration attorney.
A local immigration attorney in New York may be able to help you with local information officers at USCIS, so that you can get the necessary stamp your your passport that can allow the DMV to give you a New York State ID, sooner. Otherwise, you can try this yourself, but it can get frustrating. In general, if you have any criminal arrests or convictions, there may be other complications, delays or even Immigration Court proceedings.
I strongly recommend that you consider the assistance of an experienced immigration attorney. There are limited reasons to secure a green card replacement. If not, then you can try to file online at www.uscis.gov at your own risk. Perhaps, you can work it out and get the documentation that you need to secure a New York State ID.
The above is general information and does not create an attorney-client relationship.
Unfortunately, in these circumstances, you are unlikely to be able to get an extension through InfoPass. If you had filed for the extension already and needed a temporary stamp while the case was pending, the answer might be different - but here you are obligated to file for the extension, and InfoPass won't help you until you do.
As my colleague mentioned, you should find a way to get an attorney's assistance with this - there isn't much you will be able to do without a valid green card, and if there has been any kind of criminal problem that pops up on the fingerprint check - even one that hasn't come up before - this could get very, very complicated.
While technically you are still a permanent resident without a valid card as long as you haven't abandoned your status or done anything else to lose it, without proof of your status there could be all kinds of potential problems.
The renewal process can take three to four months. You might try, if finances are really tight, to seek the help of one of the local law school immigration clinics. You will probably still be expected to pay filing fees, though.