Can a 2013 iMac's drive be replaced to run new software?

Replacing the drive will not let you run a more recent version of macOS. Catalina is as far as any 2013 iMac can go.

Catalina is enough to run

• The current version of the Firefox Web browser
• The parts of Affinity Studio that are "free with registration." (The AI stuff that requires a paid subscription requires an Apple-Silicon-based Mac running Ventura or later.)
• LibreOffice 25.8.4. The current version (26.2.0) requires Big Sur or later, but 25.8.4 is compatible with Catalina. Right now, there is a link to 25.8.4 on the main Download page; this may eventually move off to the archive.

It is not enough to run

• Current versions of Safari, Pages, Numbers, Keynote, and GarageBand
• Current versions of most major third-party browsers other than Firefox
• Applications from vendors like Adobe and Microsoft who have a policy of supporting only the "most recent three"

If you're referring to the OS, no.

If you're referring to something else, only if its system requirements specify a higher-capacity drive; it's easier to move content to an external one instead.

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Click the apple logo top left of your Menu bar and select “about this mac” to get exact details; Then, use the following link and go to the section called “Check macOS compatibility“ against your machines info.

Find out which macOS your Mac is using - Apple Support (CA)

You can update it to the latest possible OS as per that link, And you can find the operating systems in the app store to download.