How do I determine if my laptop can charge through USB-C?

Most certainly, you won't. Although you won't get any confidence unless asking the dell support or trying it out, since I couldn't find any information regarding USB-C charging in the manual of your laptop. But here's how I come to my conclusion:

At first, if your laptop was enabled to be charged over USB type C, it most certainly would be mentioned in the manual. But the only two hits when searching the manual for "USB C" mention USB C for data transfers.

Then, Charging over USB type C is done using the USB type C Power Delivery specification. According to that specification, the maximum power that may be supplied over USB type C is 100 Watts (5 Ampere at 20 Volts). The output of your laptop's power supply is way higher than that, 180 Watts. Not surprising, regarding the hardware of your laptop. So you definetly won't be able to charge your laptop over USB type C while using it. And while it might be possible to charge your laptop over USB type C while turned off, the manufacturer probably didn't include this feature: If you can't charge your laptop using USB type C while in use, manufacturers tend to leave this feature out, even for the shut-down mode.

Furthermore, your laptop isn't listed in the list of USB type C chargeable laptops on wikipedia, which, of course, is not complete, but a good indicator.

Of course, you'll only get confidence by trying out or asking the dell support hotline. But don't be surprised to get a no.

This can be a confusing topic because the port is Thunderbolt 3, the shape of the port is USB-C, and the charging protocol is USB PD (power delivery), and Thunderbolt 3 ports are also capable of communicating via USB 3.x in addition to the much faster Thunderbolt speeds.

In this case I'm happy to let you know I also have a Precision 7510 and it can be charged via the Thunderbolt 3 port, and despite Dell's recommendation that you use a power adapter in addition to their TB15 or TB16 docks, the 130w the docks output if powered by a 240w power brick is usually sufficient to power the laptop if you aren't running 64 GB of RAM fully utilized and rendering crazy 3D models.

I have a wide variety of chargers available and in my experimenting I've found that it can actually charge from all the way down to about 65 Watts via a standard Dell charger or USB-C, but using anything less than a 65 watt USB-C typically requires your laptop to be in a suspended state otherwise it's using more power than it's receiving.

One thing to be aware of when using chargers that are less than the recommended size for your system is it will typically limit system performance due to throttling. You can override this with some applications to force "Performance mode", but in doing that your battery may discharge even though you are attached to a charger because it can use more then you are receiving as mentioned above.

Another thing to note is there are multiple voltages and amperages that are defined as Power Delivery compatible so even though you might be plugged into a charger that supports USB Power Delivery if it doesn't match what your device understands you won't get a charge. I noticed this when using a mid range USB-C power adapter that the 7510 simply ignored it even though a smaller one worked fine. Using the same adapter with a Precision 5510 or my XPS13 (9360) it works just fine.