Unless you qualify as a dependent (spouse or child) or someone who is maintaining lawful nonimmigrant status, you cannot do so.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
You cannot maintain status with a university which is not authorized to provide status form for international student. Find one that does as soon as possible and seek reinstatement.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter,not should it be viewed as establishing an attorney client relationship of any kind.
A university which doesn't issue an I-20 will not be helpful in your quest to remain in F1 status. You should attend a different institution in which this is possible. If you are out of status more than 5 months, it could become a serious problem.
This advice does not form an attorney-client relationship and is merely informative. It should not by itself be relied upon to address a legal concern.
The University must be authorized to accept foreign students and they need to issue an I20
Attorney Robert Brown's (former INS Director, 1972-99) reply to your question is general in nature, and does not constitute legal advice as all facts are known to him. For specific advice or representation you should retain an attorney experienced in immigration law. Mr. Brown's reply on AVVO does not create an attorney/client relationship not constitute legal advice.