PIP is initiated by the management or probably by your supervisor. It is a very structured monitoring process of under performed employee.
Reason is depend pretty much on the mercy of your employer or how much that troublesome employee is willing to commit to do all the necessary action item within a time frame, usually 3 months.

After 3 months the result would be reviewed, and to decide whether to terminate the employee, extend PIP or deactivate the program.

Yes, PIP sounds very demotivating at first and most of the affected employee would just opt for plan B which is to nego separation scheme (resign with dignity and a bit of compensation) with HR rather than doing the PIP, if applicable.
But if your people have all the semangat juang and confident they can achieve target as per the PIP, by all means do it.
In the end depends on the company culture, whether you believe in the system and think PIP is fair for you to fix that employee. Read all the horror stories in the internet with a pinch of salt.

If the management agree to put an employee in PIP they must agree to provide necessary tool/counseling/coaching within the time frame of the PIP, not after.
Employee must also be proactive to keep everything as record, all log books, emails, verbal communication, everything just to be sure.

Umm, i'm not sure on the details of your case since i'm not from the sales department.
But i do think the basic of the PIP is the same. Been here, done that. fought well and i survived for now...but i wont put up my hopes too high especially with the restructuring around the corner.
Your employee can always keep a personal daily record of their work if no emails/letter given. They have the right to meet the managers daily/weekly to discuss about their progress. Please record them all. The manager would feel leceh to keep up with them, but that is the way of the PIP. Both manager and employee must be proactive to fix the problem, otherwise it will backfire the manager.
This is one of the way to see which employee puts a real effort to survive...just dont give up! The manager probably will feel ashamed of himself if the employee can prove he's been doing his job & can meet his target within the time frame. Sometimes it's just only communication problem within the manager & employee. It is sad when the employee gets demotivated, no means of channeling his concerns to the management, manager sees only the weakness of the employees and BAM, marked as poor performer.

It's a long story, and it's a unique one just because i am different species in this industry which mostly dominated by men.
I had a fair share of harassment but i cant say a word just because he's a manager (very skillfull but asshole attitude).
Been transferred to different department, i was happy for a while and the cycle repeats again. I once was assigned working night shift for almost a year without any competent person. I was by myself doing all of the reports and making a lot of mistakes. But when i raised my concern nobody gives a fuck. I feel demoralized and sad, and became defensive with everyone. My latest manager not sure how to approach me & straight gives me PIP due to my last year poor review. I knew it will eventually leading to this, i dont know what to do at first. but then maybe this is the right time to prove myself that i am worthy, it's just communication breakdown. The good thing is during the 3 months i am entitled to use all the counseling benefit and travel for courses. I have to work hard both during my shift & attending courses during my off days. and of course keeping reports. No doubt the program changes me to a new person though. Well i dont really care if i made it or not in the end, at least i close the gap and they see me as trying to improve myself. Once you're out of this, this is the biggest takeaway you can do for yourself.

they dont need any signature, as long as you get from email stating your PIP will start at this date to date..then it's official.
if you have not secured anything good yet, just hang on & play their game. 3-4 months salary plus overtime & allowance and benefit is good for now rather than 'mutual separation' package which only paid basic until at the end of the year.

my company have 3 stages of PIP, first one counseling, 2nd one verbal warning & 3rd written warning for duration of 3 months. if the company is hell bent to terminate you after the last warning stage, look it this way..9 months of full salary is better but then fired maybe no compensation..

your choice.

Dont lose hope yet...just pray your HR will be fair and not siding to your manager. at least you can still buy time to look for another job.

Or try nego with HR if they will offer some sort of mutual separation instead of you just resign without any compensation if you dont want to do PIP