Can you sue a third-party for losing a vehicle?

You can only sue based on "negligence" or unreasonable carelessnss if there is evidence of carelesness, such as that the trailer was left unlocked and unattended in a dangerous location, and that's why it was stolen; without some evidence of carelessness, you cannot sue on that basis.
Similarly, you can only sue for theft (for intentionally stealing it or helping another steal it) if you have evidence of that. 
Therefore, you may have difficulty suing the person in whose possession the trailer was, without some evidence that they either carelessly or intentionally caused its loss. And for the same reason, you may have difficulty suing the person to whom you rented it unless there is evidence of their carelessness or intentional wrongdoing. The problem you have is that person A is not normally liable for the criminal act of person B unless person A deliberately assisted or helped them commit the crime, or A's carelessness caused or allowed the crime to happen. So if they are alleging that someone else stole it, if that were true, they would not be liable--and to show it's not true, you need evidence.
If the person to whom you rented it violated any terms of the rental agreement you had with them (if you had a signed agreement with them), then you may be able to sue them for breaching that agreement. Or if the agreement stated that they are responsible for any damage or loss of the trailer, you could enforce that against them to make them pay. So you ability to sue will come down to whether you have a written contract with someone and, if so, what rights does it give you.