Could I ask an employee of my company's client for a recommendation letter ? Will it violate any company policy? I will be applying to Graduate schools for Fall 2019, so far I have given my GRE and will be planning to give again, I work full time as a Team Lead at a software company and am planning to resign if I get an admit into a good school.

Now my undergrad CGPA is not very good as I had health problems and I will be probably mentioning this in my SOP.

The thing is that the company I work at has a contract with a very huge software company, lets call this company BIM (name changed) and I work for this company as a contractor but involved fully in software programming and development. The people I work with at BIM are scientists and practically the rightest kind of people to write Letters of recommendation if you want to get into a good college especially to offset the low CGPA. My work experience will probably also help with this but LOR's from scientists is a big deal.

My manager is alright with approaching them for requesting LOR's for the application process, my Project Manager is not. Apparently he believes it will create a "chaos" and probably thinks it would make the clients uneasy (the BIM people I work with might feel uneasy since I am handling a major project) 

Now, I spoke to HR whether it is against company policy and waiting for her to reply. Nevertheless I am thinking of going ahead and requesting the BIM clients directly for an LOR since I dont think in anyway that it is a violation of any policy just to ask for an LOR.

What do I do ?
HR can get you the answer concerning company policy. No one here can make a determination other than by guessing. Wait for them to respond or follow up with them. 

But also consider the consequences of making your project manager unhappy. Again, you can judge that, but no one here can.