Your transaction has
135 confirmations as of writing so it is surely enough to be credited to your account balance.
Most of platforms require 1 to 3 confirmations to credit your deposits to your account balance. First, you can check it from your side: Visit their website, find ToS (Terms of conditions/ Terms of Service) or FAQs pages, and search with deposit + confirmation to see what they find out their minimum requirement on confirmation for your deposits.
Confirmations are only thing companies need to credit your account balance. The more confirmations one transaction has (deposit for your case), the less risk of reversibility on that transaction. For your case, 135 confirmations are far from enough. Additionally, bitcoin transactions are almost irreversible (it is not profitable to do such attacks).
What is the third-party company you use? Would you mind sharing their name, website then I and others can check and help you more, please.
The third party is saying they need the wallet issuer (i.e. Electrum) to confirm the transaction in order to apply to my account.
Is it exactly what the company talked to you? It is weird. Wallet software does not confirm transaction, miners confirm it instead. Wallet software help you connecting to nodes, sign and broadcast your transaction but miners / mining pools are the ones will confirm your transaction.