It's possible to 'farm' with any toon.
It's just a matter of how fast/efficient it is. The breakeven point, in my opinion, is when you're killing fast enough to have a constant stream of the inspirations of your choice, even after combining and throwing out singletons.
SD/axe should be fine, especially if you add in reactive. It would be far from the fastest, however.
That said, an SD/axe would be much more capable in most standard content than more extreme farmers. LOTS of knockdowns in that combo. 
The regular content is a lot more interesting in any case. 
In other words, no.
Doing a respec and was wondering if anyone could help me build a SD/Axe farming spec with either arctic or pyre mastery for the epic pool.
Look at other Shield Defense builds on these forums. It isn't hard to make a good Shield Defense Tanker build. Farming builds are not much different from normal builds except with certain powersets.
The first thing a Shield Defense Tanker needs is to soft-cap defenses, which means you reach 45% on all three positions. This is easy: slot your togges, add Phalanx Fighting (no extra slots), add Combat Jumping and Weave, slot a Steadfast Protection 3% defense somewhere, and slot a full Gaussian's set in a to-hit buff power. No other IO sets should be needed for defense, which leaves you with a lot of room for recharge bonuses.
That's going to be expensive... but I was asking for help with the secondary and pool powers. Which do I take and which do I leave behind, how many slots in the powers, so on and so forthwith.
Not that expensive, at all. It's very easy to slot defensive toggles with Red Fortune, or even standard SO/IO's. If you're on Virtue, send me an email; @GI Justice, and I can provide you with some influence, or bring you on a farm. At bare minimum, I can provide enough to get to IO'd with standards.
I've been toying around with the sd/axe and been looking for some help with a build. Is it possible to farm with this set up?