The short answer is “yes”: as far as the marriage is concerned, there no problem whatsoever. The Catholic party may freely receive Communion. (The non-Catholic party may not; see Code of Canon Law, Can. 884.)
The Church makes no distinction whatsoever between marriages contracted in the “ordinary” way and those contracted by convalidation.
Hence, with the usual conditions (the person who wishes to receive Communion should not be conscious of having committed a mortal sin since his last confession), there is no restriction of any kind on receiving Communion.
As the Code of Canon Law puts it,
Any baptized person not prohibited by law [which includes practically any Catholic in good standing] can and must be admitted to holy communion (Can. 912).
(For an overview of what convalidation is, see the Code of Canon Law 1156-1160.)