Le instamos / Le animamos
 

How do I form the rest of the sentence after using le instamos or le animamos to mean We urge you to join us....... / We encourage you to join us.......

Possible constructs that come to my mind are:

Le instamos / Le animamos a que nos úna en.....
Le instamos / Le animamos a que se una a nosotros en.....
Le instamos / Le animamos a unirse a nosotros en.....

..... los juegos, las manualidades, y bla, bla, bla.

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. ;)

Thank you, AngelicaDeAlquezar. That clears it up for me. :)