Your phone must be turned on, have an internet connection, and have Pulse SMS set as the default messaging app for the app to work across different platforms.
First, try just restarting the phone. Pulse SMS relies on Google Play Services to receive push notifications on your phone from the web. This is how Pulse SMS is able to do everything instantly. Restarting the phone will restart Google Play Services and any Google-related connections.
If that doesn't solve your issue within a few minutes of the restart, go to My Account in the Pulse SMS Android app's side menu, then select the Refresh Account option. This will force you to sign in again on the web/desktop platforms. You will also need to clear the app data and sign in again on any tablets connected to your account.
The other thing that you could do is to completely delete and recreate the account from the phone. Google will restore any purchases and you will not lose any messages. On the Pulse SMS Android app, go to the side menu > My Account > Delete Account. Then, head to the Login option in the side menu to sign up/recreate the account.