Hello, I have a office subscription but I have just switched computers to version mac Mojave. It is saying I cannot download the application. Is there a way I can still use office on this version without doing a software update?

In Microsoft's list of requirements for Office 365 they state that for macOS you need:

One of the three most recent versions of macOS. When a new major version of macOS is released, the [current] macOS and the previous two versions.

This means that Office 365 cannot be installed on a 10.14 Mojave system. You would need to upgrade that Mac to macOS 10.15 Catalina to install Office 365.

Basically, what David McKinlay stated.

Even if you upgraded to Catalina at this point, once Microsoft shifts their minimum supported operating system to Big Sur in Fall 2022, software updates for Office subscriptions, or the single-purchase Office 2021 for Mac would end on Catalina — because it would have dropped off Microsoft's supported platform list.

What you might consider is a replacement Office suite for MS Office, and a good choice would be SoftMaker's Office 2021 solution. Works on macOS all the way back to High Sierra, with Windows and Linux products as well. Although there is a free trial, it is not a free product and is very likely easier to adapt to than the alternative, and free LibreOffice Suite. The other bonus is that neither of the linked alternative products would require you to upgrade from Mojave.

Did you use Migration Assistant to transfer apps, data and settings from your old computer? In this case Office should work (but you may need an activation code).