With the ongoing shutdown, it's causing some test flights of SpaceX and others to be pushed back. If it goes on for an extended period, is it possible for said rocket company to launch from another site, like in French Guyana?
No. There are at least 2 government agencies that must approve of a launch under normal circumstances, the FCC and the FAA.
Per FCC guidelines
Satellites authorized by an administration other than the United States do not require any FCC approval if earth station operations are exclusively outside the United States.
The FAA requirements are as follows
An FAA license is required for any launch or reentry, or the operation of any launch or reentry site, by U.S. citizens anywhere in the world, or by any individual or entity within the United States.
An FAA license is not required for space activities the government carries out for the government, such as some NASA or Department of Defense launches.
So you could only do it if you are launching by a rocket company that is not based in the US, and has no intention of ever communicating to an Earth Station inside of the US. A military launch, however, could still be approved, as it does not require FAA authorization.
Violation of trade restrictions
In addition to @PearsonArtPhoto ‘s answer of the FAA and FCC regulations, launching from another facility would violate the International Traffic in Arms Regulations and Export Administration Regulations ( ITAR and EAR).
From the ITAR wiki page:
For practical purposes, ITAR regulations dictate that information and material pertaining to defense and military related technologies (items listed on the U.S. Munitions List) may only be shared with U.S. Persons unless authorization from the Department of State is received or a special exemption is used. U.S. Persons (including organizations; see legal personality) can face heavy fines if they have, without authorization or the use of an exemption, provided foreign persons with access to ITAR-protected defense articles, services or technical data.
All rocket and satellite technology (and its associated technical data) is considered ITAR. You would need special permission from the State Department to export it. Which you wouldn’t get.
The government shut down did not stop SpaceX from launching an iridium satalite.
SpaceX has many existing launch licenses. See Faa.gov below.
A launch pad with a launch tower suitable for that specific rocket would be neccessary. Also a launch control center with all those computers and terminals. Antennas and receivers for the telemetry of the rocket.
A lot of connections fitting to the rocket for tanking of fluids and gases. Many electrical connections for ground power and sensors needed for all checks during count down.
All that equipment present in French Guyana fits to other rockets. There is no unified universal launch complex that may be used for many different rockets from different nations.
To build all equipment missing there would take many month at least, if not a year or more.
Construction of a new launch pad for the Soyuz rocket was started in the Guiana Space Centre 2007 and finished 2010.