I am a Mac and iOS Developer. As I understand, an active Developer account is necessary to keep certificates up to date whether your app is distributed through the App Store or privately.
Here is a descriptor of certificates from Apple's site: "You will need to set up certificates to develop and distribute Mac apps. To submit your app to the Mac App Store or distribute it outside of the Mac App Store, it must first be signed with an Apple-issued certificate. Certificates allow the system to identify who signed the app."
It sounds like one of two things has occurred. One, your company never went through the steps to get a certificate from Apple. Two, the certificate has expired.
If you're having trouble running the application issued by your company, and it's on a Mac, you can go to System Preferences > Security > General. Click the lock-symbol in the lower left and input your password to unlock these settings. You can then change the security to "Anywhere" while you run the app the first time. Make sure you switch it back as this will remind you in the future if you have downloaded something from the internet that might have questionable origins.
You might also want to bring this to your company's developer. They can submit a request for a new certificate so this issue does not continue.