Yes, you can apply for EI as soon as you are let go, and your employer will send your Record of Employment directly to Service Canada so you do not need to submit it in person.
However, there are two major things to think about with this answer. First, EI will not pay any money while you are receiving severance payments from your employer. Even if you agree to a lump sum settlement that represents X number of months, you cannot start collecting EI until that time period has run out.
If you are still negotiating, and were hoping to collect EI in the meantime, speak with us first. We understand that you may be dependent on having a source of income to cover your necessities, but collecting EI can add a complicated wrinkle to your settlement negotiations, and possibly delay your payout depending on how the employer decides to proceed. You can start the application process early, but speak with us before you make any claims.