Keep in mind that you need to stock said ship with reasonably competent characters. If you have Sups 1 & 4, the NPC's therein can be quite useful for doing this quickly.
Otherwise, yes, it's possible to claim as prize, but no, it's probably NOT legal to actually KEEP the ship. Traveller mentions this in a few places, but they're out of the way. So, look to real world praxis...
IIRC, Current typical real world runs between 1% and 10% of fair condition value upon surrender to the admiralty courts. Often, if the pirate isn't a dedicated warship, the capturing vessel typically gets the opportunity to buy it at a reduced cost from fair value.
10% of a MCr40 ship is still MCr4 in cash from the admiralty...
But also, fair value is typically 90%-(10% per decade), and then is reduced further for the cost of repairing the damage. (While this is somewhere in CT or JTAS, I don't recall where; I grabbed the numbers from T20 THB page 279.
See that this is today.
I'm not a historian, nor a lawyer, but I think in the Age of Sail (which Traveller depicts in some senses) any pirate ship captured was fair prize, (unless captured by a Navy, in which case it became Navy property).
So, that would depend on the setting, or even the interpretation you make of it (mainly in the case of the OTU).
I'd suggest you to read (if you have access to it) the Challenge article
Prize Court (issue#38, pages 25.27 and 79).
Basically, if a ship is captured by a naval crew, it becomes Imperial property that uses to sell it at auction (the capturing crew reciving, in theory, about 30% share).
I captured by a privateer (and I guess registred Starmercs), the crew receives about 25% of the selling prize, and the ship owner about 50%. As the ship is likely to be old and surely it is damaged. the price can be quite (relatively) low, though.
OOH, this probably means that the privateer is able to keep the ship by paying the 25% sahre to the government issuing the Letter of Marque...
This does not mean this is easy, though. As I would apply it:
First of all, it must be proved that there is no legitimate owner for the prize ship (and this can take several weeks at least), as a pirate ship is likely to be a prize with a legitimate owner claiming for it (even if now the legitimate owner is the financial company that paíd the insurance to the former owner). If so, the capturing crew is likely to receive a bonus (let's say about 10%, as Aramis said above) of the ship price...
If, after some time, it is confirmed that there is no legitiamete owner (or it is an enemy of the state), then the auction will take place, with the capturing crew receiving its fair share (though the selling price is likely to be quite under the nominal one).
For privateers and starmercs, I'd give them the right to buy it on a first refusal basis, by a 25% (the usual government's share) of its current (damaged and old as it is) price. For the government, this will probably represent a better share tan if sold at auction on the prize courts (where the same article explains that corruption and agreed prices among the larger companies are the norm), while also keeping happy its privateers and starmercs, and augmenting those auxiliary forces.
For any merchant occasionally taking a pirate ship as a prize in self-defense, I'd treat them as privateers and starmercs for this matter (as long as they can prove they were not in fact searching for a prize, that would make them probably close to piracy...).
Then, of course, the ship must be repaired and refitted before it comes to any use...
As you see, posible, but neither easy nor cheap...