Yes, as you've noticed, some parishes do offer a Service of the Word with distribution of Holy Communion from time to time.
The conditions for conducting such services are described in this 1988 document:
. . . issued by the Holy See, from the Congregation for Divine Worship, the Vatican department in charge of liturgy.
These services are regulated by the diocesan bishop, and the US bishops' conference has issued a book of guidance for Sunday celebrations in the absence of a priest; these may be led by a deacon or (lacking that) an authorized layman.
In the U.S. at least, they can also be offered on a weekday if the parish has no scheduled Mass on that day, though the procedures in that Sunday book would not all apply.
Some guidance on weekday celebrations in the absence of a priest is online at the US bishops' web site:
I hope this helps!
With best regards,