To my mind, if you're trying to send an invitation, "instar" is not a friendly verb, as it means to exert pressure on someone to do something, either because of some authority over them or because you're insisting, or rather pushing someone to do what you want. If the intention is that people join some compulsory activities, then "le instamos a tomar parte de..." would be alright.
However, for activities like the ones you describe, which are normally noncompulsory, "animar" is fine.
Your sentence "le animamos a unirse a (las actividades)" is correct and there is nothing wrong with it, so you can use it without worrying.
If you want a more "Mexican" sentence, you may say something like: "Le invitamos a tomar parte en nuestras actividades como juegos, manualidades...".
"...formar parte de..." is also valid, but I like "...tomar parte en..." better because it suggests a more active attitude.
