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.
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I-797 for based on what kind of application?
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The answer to this question depends on facts you failed to include in your post
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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.