It's not your plant?

Yes, you could stake it.....
but, truly, there are other issues at hand: the potting mix is too moisture-retentive, and the pot is too large for these plants in this mix. I imagine that the plants also need more light.

Josh

It has already been staked....in the heart. Joking aside, Josh hit the nail on the head.....in those conditions, it will perish.

Joe

Try a grit mix or add %50 cacti succulent soil with 50% perlite for drainage slowly introduce to strong Sun light and he'll perk up

Since it's not your plant, maybe just stake it for now & wait to show it to the Owner. But pls. whatever you do, do not water it further or might well send it over the edge.

Josh's comments are on the mark & he really knows these plants.

Depending on how long you're house sitting, the Homeowner might have told you not to water that plant for several weeks (unless they're new to plants as well).

That tiny, urgent sound you hear is that poor plant screaming for more light.

Great advice - this plant is next to a window but the window looks out onto a porch so it is definitely not receiving tons of light. I have only watered it sparingly twice in the two and a half weeks I've been here, but perhaps that is too much?

I was given instructions to only water if it was dry, so I've just been testing the top soil. Perhaps I need to check deeper?

A little intimidated about re-potting it - however, the owner will not be home until July 3rd though and I don't want it to get any worse!

And I am right that this is a Jade? I identified it by looking at pics online so I"m not sure.

Here is my attempt at scaffolding...

Personally, I would not repot someone else's plant w/out their permission.

Stakes are OK. I suggest you wait a week to water again.

These plants can recover from underwatering, but not overwatering, so we usually recommend to new succulent growers that when in doubt about watering, don't.

If you skipped watering for the rest of month 'til they get home, the plant would still be OK (they hold water in their fat, fleshy stems).