How do I stop others from accessing the “play sound” button on find my device?

It is not possible to disable the Play Sound function.

However, it is only possible to do this from devices that are signed in to your Apple ID. So you can ensure that your family does not have access to iPhone, iPad and Macs that are signed in with your Apple ID. You can also change the passcode for those devices so they don't have access.

Alternatively, you can set up Guided Access, on iPhone and iPad. With Guided Access, you can prevent your family from using apps other than the one they have open. For example, if they are playing a game, then with Guided Access you can prevent this game from being left.

You will find Guided Access here:

Settings → Accessibility → Guided Access

You can find more information about Guided Access here:

Use Guided Access with iPhone, iPad, and iPod touch - Apple Support

Okay, if you use Family Sharing, that feature cannot be deactivated, even if you choose not to share your location, they will still be able to "help" find a lost device, for example by playing a sound. I think the only solution is a good conversation with your family.

If you don't share your location, family members can't access location information about your device. However, family members can still help you. They can check your missing device's online or offline status, play a sound on the device, put the device in Lost Mode, or remotely erase the device.

Source: Apple Support.

You can find more information here:

Share your location with your family - Apple Support

Does your family use the same Apple ID as you, or do they each have their own Apple ID and do you use Family Sharing?