Is there a RAW way to summon a monster of some variety and make it permanent? In my scenario I'm thinking of Oozes specifically, in case that makes a difference.
Not in a straightforward way
There does not appear a clean way to do this. (To be honest, even to summon an ooze in the first place seems to be difficult, as oozes are not one of the options of summon monster.) Here are a few ways that might work, with a lenient DM:
This would only work on an extraplanar ooze, and one that is intelligent to agree to a trade to boot. Not a great selection there. Gate allows you to call any kind of creature. It says:
The second effect of the gate spell is to call an extraplanar creature to your aid (a calling effect). By naming a particular being or kind of being as you cast the spell, you cause the gate to open in the immediate vicinity of the desired creature and pull the subject through, willing or unwilling.
To get the called creature to stick around permanently (or at least, for a really long time that approximates permanently), you then need to do this:
If you choose to exact a longer or more involved form of service from a called creature, you must offer some fair trade in return for that service.
I'm not sure how the two effects interact, but possibly binding your summoned creature in a hedged prison could keep it from disappearing. It says:
The subject is transported to or otherwise brought within a confined area from which it cannot wander by any means. The effect is permanent. [...] A bound extraplanar creature cannot be sent back to its home plane due to dismissal, banishment, or a similar effect.
So this would come down to if you treat the expiry of the original summoning as a "similar effect" to dismissal, as it sends the creature back to its home plane when the spell ends. I probably wouldn't.
Wish is not as powerful as in older or newer versions of the game, but after a list of things that does not include making spells permanent, it still has this language
You may try to use a wish to produce greater effects than these, but doing so is dangerous. (The wish may pervert your intent into a literal but undesirable fulfillment or only a partial fulfillment.)
So in theory this could work, if the DM plays along.
Polymorph any Object
This is not summoning in the sense of calling from a different place, but it is calling forth an Ooze nonetheless. You need a creature of higher intelligence and of the desired size. Polymorph any object then can turn it permanently into an Ooze (as you get +5 for same kingdom, +2 for same size, and +2 for lower intelligence, and 9+ is sufficient to create a permanent effect).
Get a mundane Ooze
This is not a direct solution to the problem of having a summoned Ooze made permanent, but if your goal is to just have an ooze for your dungeon or keep, a simpler way to get one may be to just capture one and put it there. The good thing about this one is that it is clean-cut by the rules, so needs not a lot of DM buy-in.
For example, you could baleful polymorph one into a mouse, pocket it, travel to your place, and dispel the polymorph.