Voltage drop can be several things.
First, it can simply be the length of the circuit. You are reporting 12 volts of voltage drop (10%). If the circuit is wired with 12 AWG wire and is 290 feet from the service panel to the outlet, then that voltage drop is entirely explained by the length of the circuit, and you need look no further. Come talk to us before buying any larger wire for that run; we can save you money (versus using costly #8 or #6 copper) and get you into level 2 charging too! Win/win.
The second thing to look at, if it's a SMALL business, is a Lost Neutral at the service. If the Tesla charging causes some circuits in the business to drop to ~108V but others to pop up to ~132V, that means the service neutral from the utility has been lost. The voltage drop is due to neutral pathing through the ground rods and dirt. I would be expect it to be more extreme though.
If we cross off the first two, then we have a big problem. That kind of voltage drop, happening at a single point, is destructive. It's being lost to heat at a single point in the system - 12 volts x 12 amps = 144 watts being applied to some splice or junction that can't handle getting that hot! So a careful search of all the wiring in the chain must be done to hunt down the fault. Don't run the charger for more than a few seconds for diagnostic purposes. This can start a fire.
I'd like to bring in a multimeter to prove it's low, as I don't think anyone is going to believe my Tesla, but I'm not sure how prove it with a multimeter. If I plug in the Tesla into the upper socket, and the multimeter into the lower socket, should that show the "voltage under load"?
That depends. There's a small chance the socket is wired as a MWBC (Multi-Wire Branch Circuit) aka shared neutral (2 hots 1 neutral). To check that, look at the 2 sockets, look at the hot pin of each socket (the shorter pin that is NOT T-shaped), and measure between the hot pins. If you see 0V it is a normal outlet. If you see 208V or 240V it is wired as MWBC.
So what? If it is wired as MWBC, "plug EVSE into one socket and measure the other" will give a weird reading. Voltage on the measured side will actually increase slightly. Assuming you have consistent voltage drop (6 volts on hot and 6 volts on neutral) the second socket would read 126V.
Actually, you can take hot-ground and neutral-ground measurements to be sure. If 12V voltage drop is simple wire loss, then you will see neutral-ground voltage of 6 volts and hot-ground voltage of 114V (6V less than when the car is not charging). If the voltage drops are equal on each wire like that, that's a reassuring sign that it's not an impending wire failure. If the voltage drop is mostly on one wire, that is a failing connection - kill it with fire before it kills you with fire! (or more accurately, gets you disinvited from charging at work lol).
Remember that if you can talk your office into installing Level 2 charging, two important facts: #1 Tesla Wall Connectors have a "Power Sharing" feature where a bunch of EVSEs can dynamically share a single current allocation. So you can put in eight Wall Connectors and have them share, say, 60A. "1440 watts per car? What good is that?" Except every time a car finishes, its allocation gets spread to the other cars. By afternoon only 1-2 cars are still charging and they get plenty. It's much better than EV owners dragging other EV owners out of meetings to move their car coz I gotta charge. And #2, Tesla makes a J1772 Wall Connector which Power Shares with the others, so that every other EV that exists on the continent can also charge. Tesla makes a quality adapter for you to charge off J1772; the reverse adapters are cheap Cheese from overseas and unwise to use IMO.