Why sleep stops checking
Checking is not done by a server somewhere. The Chrome extension opens your watched Upwork search in your own browser, using the session you are already signed in with, reads the results, and sends them to the app. That is what keeps it inside the rules Upwork sets for a person browsing their own account.
The cost of that design is simple: when the computer sleeps, the browser stops, and nothing is checked until it wakes. Turning off the screen is fine. Locking the screen is fine. Sleep and hibernate are not.
What it looks like when it happens
The job alert screen says checking has not run recently, and names a sleeping computer as the usual cause. It only says that during the hours you set for checking, so a closed laptop overnight or at the weekend is never reported as a fault.
Nothing is lost while the machine is asleep. When it wakes, the extension runs the check that was due, sends anything it could not send earlier, and the inbox fills as normal. Jobs posted during the gap still appear, because the search page still lists them.
Windows
Open Settings with Windows key + I, then System, then Power & battery. Under screen and sleep timeouts, set the sleep option to Never for the plugged-in case, and leave the screen timeout at five or ten minutes.
There are separate values for battery and mains power. Setting Never on battery will flatten a laptop, so change the plugged-in one and leave battery alone.
If you prefer the command line, powercfg.exe /change standby-timeout-ac 0 does the same thing, and powercfg.exe /change monitor-timeout-ac 5 turns the screen off after five minutes.
macOS
Open System Settings, then Lock Screen, and set the display to turn off after a few minutes. Then open Battery (on a laptop) or Energy (on a desktop) and turn on the option to prevent automatic sleeping when the display is off, on power adapter.
A closed lid still sleeps a MacBook unless it is connected to an external display. If the lid is shut, treat checking as paused.
Linux
On GNOME, open Settings, then Power, and set Automatic Suspend to off while plugged in. Blank Screen can stay at a few minutes.
From a terminal, systemd's answer is: sudo systemctl mask sleep.target suspend.target. Unmask them when you want normal sleep back.
ChromeOS
Open Settings, then Device, then Power, and set the behaviour while charging to Keep display on.
A closed lid sleeps a Chromebook. There is a setting to stay awake with the lid shut only on some devices, so plan on leaving it open.
If you would rather not change anything
You do not have to. The product is built for someone working at their desk anyway: checking runs while you are there, and the gap while you are not is usually a few hours rather than days.
Two habits recover most of it. Leave the browser open when you step away for an hour, and open the job alert screen when you sit back down — the first check after a wake happens within a few minutes.