I honestly can't recall a single reference to the Reiksguard being secular - they are a religious, Sigmarite, templar order in every piece of fluff I have seen.
And beyond the single line that began appearing in the 6th edition book nothing they do makes them particularly religious. They don't appear in WHFRP Tome of Salvation that has blurbs on all the Empire's major Templar Orders and their relation to their respective cults. The novel Reiksguard sets them up a formation designed to further the interest of the Reikland Electors and power of the Emperor. They are a means to keeping Imperial unity and why sons and heirs of important Nobles from other provinces for a time serve in its ranks.
Army Book fluff is some of the worst stuff GW puts out and it keeps get and less and less sensible. For example look at this gem that is in the 8th edition book:
To become a member of one of the Empire's select brotherhoods of Knightly Orders a noble must forsake his feudal responsibilities and rights of inheritance and then pass a rigorous right of initiation.
That is of course ludicrous. Certainly a number of Templar orders which have religious vows would require initiates to cast off their secular responsibilities, the context of the sentence is that all Knightly Orders must do so. In the previous book this stupid line wasn't even there but simply that nobles of the Empire often join Knightly Orders.
There is no canon to Warhammer background and you can place primacy of fluff on the increasingly dreadful Army Books if you like but they are far from the best source.
Current army book, page 44, second last paragraph talks about them being drawn from every state and only requirements being loyalty to emperor and martial skill. No where on the page does it mention anything about being a religious order.
It still includes the copy and pasted line from the 6th edition in the first column calling them 'Devout templars of Sigmar'. Of course the whole Reiksguard page in the new book is a frankensteined copy and paste from the 6th/4th/ and Enemy Within with slight alterations and fraught with contradictions from when compared to the previous page on Knightly Orders, the quote I already included about having to forgo inheritance yet specifically stating that many Reiksguard knights are heirs to Electoral seats.