No, you will not get Savage queues to pop in Duty Finder—Savage is not designed for Duty Finder.
If you have the floor unlocked, you can make a Party Finder for it.
Which floors are you looking to do? After you run through the normal modes of both Deltascape and Sigmascape, you can accept a subsequent quest from the little Magitek terminal in Rhalgr’s to unlock the Savage versions of both tiers. Since those are both older tiers, the floors are all automatically unlocked for you, and you no longer have to progress and clear them in order to get to the next one. That only applies to Alphascape Savage, which is the current tier.
If you are looking to get a i360 Mendacity weapon, you can do Sigmascape V4.0 (Normal) for 4 Sigmascape Crystalloids. 4 of those Cyrstalloids + 1,000 Mendacity will give you the i360 weapon, and there is no weekly restriction on the Crystalloids, so you can just do the floor 4 times in a row. Run Rabanastre and Ridorana once each for their respective coins and trade them in to one of the vendors in Rhalgr’s for the Solvent to upgrade it to i370.
But you should honestly look into buying either an i380 crafted weapon, or clearing Suzaku Ex for an i385 weapon if you are looking to do current Savage—Mendacity weapons are outdated, and they will hinder your progression through the last two floors (O9S and O10S are easily clearable with last tier’s BiS).
There are no restrictions on farming Mendacity for i360 and farming Rabanastre and Ridorana for their respective coins to trade in for the Polish (right-side and belt upgrade) and the Weave (left-side upgrade), so you can get i370 with enough tomestone grinding for it. Or you can purchase i380 HQ crafted gear. You should already be spending Genesis on i390 gear for your main class that you are looking to raid with.
Gearing up is relatively simple, but you have to be willing to put in the time for it. Sigmascape Savage is not the only place to get i370 gear anymore.
You cannot join those PFs because the owner has specifically set it up for only players that have previously cleared have the ability to join. If you want to clear any content, you need to either make your own Party Finder with no prior clear restriction, or search for one that is labeled [Duty Completion]. The [Duty Complete] tag is explicitly for farming content, not for learning and first-time clears.
It is not “impossible” to complete content that you haven’t cleared before; you just need to not be trying to join parties designed for farming to do so.
I also want to stress something else: just know that Savage content is several steps above anything else in the game, so clearing some of it for the first may take much longer than a one-and-done like most other content. It is also important to keep in mind that the older floors do not see much activity once a newer tier is released, save for usually the last floor (O4S, O8S) for mount farming. Again, if you are seriously considering venturing into the current tier (Alphascape Savage), I would focus more on farming up i370 for both left and right side outside of Sigmascape Savage if you do not wish to invest in i380 HQ crafted and focus on clearing Suzaku Ex for your i385 weapon as opposed to farming the older tiers for gear/gear upgrades.