Access to LCR is dependent on standard callings and there is no standard calling of magazine rep. Magazine Rep, as a custom calling, will not be granted access to LCR and there is no way to give a custom calling access. The ward clerk can see all the standard callings by looking at the list of callings in LCR > Callings > Callings by Organization > All Organizations.
Someone with access to LCR > Reports > Magazine Subscriptions will have to provide a printed copy. That report will need to be provided at least monthly, or more often, since the information there is highly volatile. The magazine rep will not be able to order for a member or household as do unit leaders (bishopric, exec secretaries, clerks, organization presidencies).
There is information about the magazine subscription program in the article
Church Magazines at No Cost FAQ. That would be information the magazine rep should become familiar so he can help families and members get subscriptions, if they desire.