Update 5-12-17 -- Called LSP HQ today and spoke with Wyndie (Office Manager according to their website). My wife's was still showing Pending Review after being accepted and paid for. Mine was still showing Pending Acceptance 2 weeks after submitting. She informed me that my wife's was awaiting results of a background check and should be released and mailed within the next week. She accepted mine while I was on the phone, informed me that I would be receiving an e-mail informing me of that, but no link to pay as it is free for Veterans. I had been led to believe that the fee for Veterans was half-price and not free, but was told that is only for Active Duty, Veterans are indeed FREE for both the 5-year and lifetime permits. I opted for the 5-year simply for the added benefit of not having to do a background check when purchasing a firearm.
Wife and I both received our renewal permits 8 weeks after submitting. FWIW, they both came with updated pics from our driver's licenses, instead of the passport style pics used on our original CHPs.