I somewhat disagree with Kate's answer about waiting until after the vetting process, but I do agree that you don't want to appear to be speaking for your company if and when you do reach out.
The reason I'm disagreeing with waiting until the vetting process is completed is that the feedback you might get from the vendor can be a meaningful consideration for the vetting process itself.
Consider if they respond that they're happy with your feedback and e.g. already have some fixes to that effect; versus if they shut you down and show no interest in listening to their user base. This would very much impact my decision (as the company) as to whether I should adopt their product into my company's ecosystem.
I understand full well that as an "I am recommending this personally" means they have no reason to do it
I wouldn't by default assume that they'd interpret your feedback as an authoritative request for work. They are the vendor. They build the product. You (and your company) are end users. It's on them to decide whether or not they care about and/or action your feedback.
and my company wouldn't pay for the development time.
In the scenario I listed above, this shouldn't be a default assumption either. However, there are cases here where vendors offer products that are bespoke for a specific customer, whose wishes are used as the source of future work (and potentially billed to the customer).
But if that is the case, then your feedback is very much warranted as this will be part of the ongoing interaction with the vendor going forward. This makes it even more important that you understand if and how they respond to your feedback as part of the vetting process for this vendor.
Is it even a good idea to provide that kind of feedback in the position I'm in?
Reaching out to the vendor on your own, not talking to your employer, and risking coming across as if you are speaking for your employer? Bad idea.
But you can reach out to your employer and see how they intend to tackle this feedback. Since they are in a vetting process, they will obviously be in a data collection phase where they want to understand how the tool is perceived by its end users, and your question implies that you are such a pilot user.
To be honest, this is the confusing part of your question to me. You are a pilot user, part of a vetting process; and you're asking internet strangers how to relay your feedback? A pilot program should very much have a feedback system built into it, by the very premise of it being a pilot program.
The above assumes that you don't have a direct line of communication to the vendor. However, I've worked in plenty of scenarios where there is a direct connection (e.g. a Slack bridge channel) specifically for users and devs to communicate and address questions/issues, without having to go through multiple middle men (i.e. management, sales reps, ...) and have details of the question be lost in the process.
In the scenario that you do have direct contact and are clearly permitted to engage in that contact, simply ensure that you are asking a question, not making a demand. Consider phrasings like:
Hey, I've noticed that [feature] doesn't allow me to [behavior]. Is that an intentional design decision? Are there any improvements on the roadmap? Am I maybe misunderstanding how it's intended to be used?
This doesn't imply that you are instructing the vendor to do something, let alone that you are promising to pay them for acting on that question.