Hi Luke,
Thank you for querying in this forum.
From your description, it seems that you want to use eyedropper tool in Word for Windows, however, you cannot find this option.
May I know if my understanding is right? If so, we have tested it on our side, we went to Word > inserted a shape > clicked Format or Shape Format > selected Fill, we cannot see eyedropper option. We have also tested it with the latest version of different channels, and we found that this option is not available in Word. And this option is available in PowerPoint, for more details, you can refer to this article: Use Eyedropper to match colors on your slide
We’re afraid that there is no out of box way to achieve your requirement Microsoft 365 for Word. We’re sorry that it doesn’t meet your requirement. We do understand the inconvenience caused and apologize for it.
So to address your concern about the situation, we’d suggest you go to Word > File > Feedback to provide your feedback. This feedback will directly go to them so that our related team can hear your voice on this feature directly.
For a workaround, if you just want to know the color of shapes, you can select the shape > Shape Format > More Fill colors > you will see RGB color under Custom. As shown below:

Your understanding and patience will be highly appreciated. I hope that you are keeping safe and well!
Best Regards,
Sukie