Would the installation of a Tent being installed in a parking lot be considered commercial and allowable under FAR 13.5?
First of all, you should determine if the requirement is a commercial item. FAR subpart 13.5 is “Simplified Procedures for Certain Commercial Items”. It details general conditions and special documentation to invoke the higher SAT and the use of FAR part 12 commercial procedures. Is a tent used as an alternate care facility commercial? I would say yes, as there non-government equals in the commercial world “customarily used by the general public” per the commercial definition in FAR 2.101. Therefore, you can apply FAR subpart 13.5, assuming you comply with any of the other requirements listed there (J&A, commercial item determination documentation, etc.)
Second, you should document the urgency of the requirement per FAR 12.102(f)(1). FAR part 18 Emergency Acquisitions could be applicable, too (FAR 18.104). If you need a J&A for the sole source, I would use FAR 6.302-2 Unusual and compelling urgency.
You may want to check DFARS 212.102(a)(1) if an official commercial item determination applies (over $1M). Also check DFARS PGI 212.102(a)(1). I don’t think these apply if you can justify using FAR 12.102(f)(1) as stated above. Regardless, you will need to document your contract file stating this is commercial item per FAR 13.501(a) and (b).
Third, I suggest looking at the DPC website for new Class Deviations (CD) at https://www.acq.osd.mil/DPAP/DARS/class_deviations.html for different dollar values and locations, as some were just added that may apply to your requirement (you didn't state location or $). Note that the older CD 2018-O0018 raised the SAT to $750 CONUS. See https://www.acq.osd.mil/DPAP/policy/policyvault/USA002260-18-DPC.pdf
Note: you may also want to refer to DFARS 215.371-2 Promote Competition, since you only received one offer. But I think the exception applies in DFARS 215.371-4(a)(2).
Finally, even though this is urgent, I suggest you run this by your policy, legal and supervisor for a very quick look. If you have good documentation, then you should be fine! Good luck and thanks for supporting the front line!