Yes. For the LMIA process, it doesn't matter whether you worked/currently work for the employer and on what type of work permit. A positive LMIA decision just establishes that there is a position in the company for which a suitable Canadian/PR could not be found despite sufficient recruitment efforts (consistent with the TFWP advertising requirements), and therefore the employer is justified in offering the position to a foreign national who meets the requirements of the position. If the employer is able to demonstrate this and get a positive LMIA, you will then have to apply for a closed work permit in which you need to show that your qualifications meet the requirements of the position identified in the LMIA. So your previous or current work permit type is not relevant to this (just your education, experience, skills etc. are relevant).
Editing to add that one main issue you should look at beforehand is whether your employer is paying you at least the median wage for your NOC in the region where you work (the median wages are published on Job Bank for every region and every NOC). If they're paying you lower than this wage, then they will need to increase your wage to the required median wage and also list this revised wage on their advertisements / recruitment efforts. The logic behind this is that they can't just say "we couldn't find a Canadian" while offering a low wage that is not suitable for the NOC and the region (and so qualified Canadians/PRs didn't bother to apply due to the lower than appropriate wage). So basically it all depends on how valuable you are to your employer and if they're willing to go through the hassle to keep you employed in the position instead of hiring a Canadian/PR.
It's not prohibited in any law/regulation/policy/guideline and it's a frequent occurrence for companies to obtain LMIAs for their existing employees who are on open work permits. In Section 203 of IRPR, which governs the hiring of foreign nationals via LMIAs, there is no exclusion for foreign nationals already working for the employer or already in possession of a different work permit. This answer on IRCC's website tangentially talks about this, and based on the answer ("So in that case, your employer may want to get an LMIA"), it's clear that there is no issue for someone with a PGWP having their employer get an LMIA for them. Since you mentioned your employer already applies for and gets LMIAs, if they're doing this through a consultant, the consultant would readily confirm this as well.
Global Talent Stream (GTS) has some additional/different requirements compared to a regular high wage or low wage LMIA. But if your position and salary qualify for GTS, and your employer already meets other requirements for GTS (such as a labour market benefits plan etc.), then GTS could be an option - you would need to discuss this with whoever at your company / at your company's consultant firm works on the GTS LMIAs to determine if your position/salary qualifies. But chances are your position / salary doesn't qualify for GTS, so a regular LMIA would likely be needed (but this is not because you're working for them on a PGWP; it's because your position's pay isn't sufficiently high or the skills required aren't sufficiently unique/specialized). GTS is exempt from advertising requirements, so employers prefer it if they're hiring for a specialized and well-paid position.
