I'm worried about my child being able to chat to unknown players over the internet while they battle. Can I place age restrictions on certain features?

Yes. Both the Nintendo Switch and mobile versions have the following restrictions for players who are aged 15 or younger.
*Age restrictions may vary depending on region of residence.

・Text Chat (excluding template text)
・Voice Chat
*Chat cannot be sent or received.
・In-game purchase limits of ¥12,000 per month (separate limits for each version).
・Push Notifications
・Linking to social media (Twitter/Facebook)

For the mobile version, players aged 15 year or younger may only use a Nintendo Account or Pokémon Trainer Club account when linking an account to share Pokémon UNITE save data.