Without knowing what type of application this I-797 was issued, I can almost safely advise you that yes, you can cross immigration or border patrol check points with it, showing that you are in some sort of immigration or application process to either extend or change your status.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
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If your I-797 extended an existing status or it changed your staus from one nonimmigrant category to another you should be able to go through checkpoints without much trouble.
If the I-797 does not fall into the criteria above then it may not be sfe to use the I-797.For example if you simply applied to have a new status and you indicated you would get your visa abroad and selected consular notification the I-797 probably would not be an appropriate document tu use to go through a checkpoint.