Is there a ROM with Windows 10 Mobile for the Lumia 820? Because on lumiafirmware.com, the only ROM is WP 8.1.
There's no ROM specifically for the 820, no. However, there is a known process for unlocking the bootloader on the first-gen Lumia WP8 devices, which lets you install a customized ROM; you can use this to install one that lets you change the device identification such that it tells Microsoft it is eligible for W10M (or, in theory, you could make a W10M ROM for the phone yourself, and flash that, but it's probably easier to do the upgrade route). This process is explained in @Indrek's link in a comment under your question.
The other option is to use apps that change your phone's identification string (used by Microsoft to determine what updates you get) on your device's current, stock ROM. This is arguably safer, as there are fewer steps to mess up and most of them are less likely to leave your phone in an unusable state. They require an SD card for installation on WP8.1, however.
For WP8.1, I believe the best tool released was called
vcREG and is available on the XDA-Developers forum. The registry key you'll need to change is
HKEY_LOCAL_MACHINE\SYSTEM\Platform\DeviceTargetingInfo. You can try using the following settings, which are from a Lumia 950 XL and are eligible for W10M upgrades (but then, they came with W10M from the start, so they might not work for a device that didn't; I haven't tried upgrading a WP8.1 device in years):
"PhoneModelName"="Lumia 950 XL"
MAKE BACKUPS OF THE VALUES YOU ARE OVERWRITING, AND RESTORE THEM AFTER UPDATING!