Is it possible to travel to the Netherlands without the Schengen visa, my non-EU wife and myself UK/EU citizen? It's just for a 3 day trip.
Regardless of the length of the trip, it is only possible if your wife has a "residence card of a family member of an EU citizen." She will not normally have this unless you live in some EU country other than the UK, or if you lived together elsewhere in the EU before moving to the UK. Normally, the spouse of a UK citizen living in the UK or outside the EU won't have this.
Instead using UK issued marriage certificate, non-EU spouse's British Visa Card (which states it is a spouse visa) and husband's British Passport?
I don't know what you mean by "British visa card," but it is almost certainly not the required "residence card of a family member of an EU citizen."
The Netherlands use VFS Global for Schengen visa processing and they do not appear to have appointments for the next 3 months, or their online booking system is broken.
As the family member of an EU citizen, your wife should be able to apply for her visa directly at the consulate, so as to avoid VFS Global's processing fees, because the visa is supposed to be free of charge. Information is available at the VFS site for UK applications, though they don't mention the option of applying without using their services. This could mean that the Dutch government covers the service fee for these applications, in which case you won't likely be able to get through to the consulate.
However, you still should be able to, because the visa application process is supposed to be simplified and quick. If you're unable to make an appointment at VFS, that should also justify an application at the consulate or embassy. If you live in the UK, you can try getting in touch with the Dutch Embassy in London.