My situation is like this:

I'll be working in Company B as hiring-Company A's contractor. Can Company B sponsor H1B for me?

 I'll be working as contractor to Company B only but company A won't be sponsoring H1B . Again company B is willing to sponsor but won't take me as their employee immediately. Company B is willing to take me as an employee only after Oct. 

Please let me know if this situation can be possible or if there will be any issues that might arise.

Thank you

Its clear company A wont be sponsoring H1 for you. Also I am pretty sure company B wont be sponsoring H1 either (immediately)as it is a temp to hire contractor position. Once u get full time in company B you might get H1 sponsorship from company B. so for now start working for company B and fig out if there is chances of getting h1 from them, chances of getting rid of company A by getting full time in company B.

your situation is like you got job at ABC company(company B) through Robert half technology(company A) company A wont sponsor h1 and dont know abt company B right away. nothing to loose as of right now as u dont have H1 yet