Yes, the "competent observer" is a required crewmember in that situation, by definition.
According to 14 CFR 1.1:
Crewmember means a person assigned to perform duty in an aircraft during flight time.
Clearly, a "competent observer" operating as such under §91.109 (c)(2) is a person assigned to perform a duty in an aircraft during flight time; such an observer is therefore a crewmember. The operator is required to operate the aircraft with that observer performing their observation duties, and that person is therefore a required crewmember.
There are no other requirements as to who this person may or may not be other than the qualifying term "competent". The aircraft operator bears the burden of determining the observer's competency. One might argue that a certified pilot would be a competent observer.
An implied question here might be, is this competent observer a required crewmember that is able to log time for the performance of their duties? To this I would say, no, there is no provision in 14 CFR 61 for the logging of flight time for a person designated as an "observer", competent or otherwise.
EDIT to further explain the above paragraph in consideration of discussion in the comments: 14 CFR 61.51 allows pilots to log PIC and SIC time for flight in, among other conditions, "aircraft for which more than one pilot is required under the [...] regulations under which the flight is conducted". The "qualified observer" is indeed required in the situation in question, but need not be a pilot, and—unless otherwise qualified—is not acting as one of the required pilots in an "aircraft for which more than one pilot is required under the [...] regulations under which the flight is conducted". Obviously, if the person fulfilling the duties of a "competent observer" is otherwise also a required pilot, then the pilot may log the flight time as the required pilot.