Hello,
As a first step, try to refresh the Word app install by offloading and reinstalling it.
Before you start, open the document in Word and make sure it is saved to OneDrive or the Files app. If anything is only On My iPhone, tap the file name and move or save a copy to OneDrive or Files so nothing is at risk.
On your iPhone, go to Settings > General > iPhone Storage > Word. Tap Offload App, then tap Reinstall App. If Offload App is not shown, delete Word from the Home Screen, then reinstall Microsoft Word from the App Store, and sign back in.
Open a simple test document in Word, tap the three dots at the top right, then tap Print. The system print dialog with your printer, copies, and options should appear.
If the dialog still does not appear after this reinstall, tell me your iPhone model, iOS version, Word version, and whether your account is personal or work or school, and we will go deeper.