Can I update Minecraft and not lose modded items and progress?
Solution 1:
You would want to update both the mods you're using and update the core game itself before loading your world. It also never hurts to make a backup of your world.
Once you update your mods to support 1.13 and update the game itself, load a new world to make sure you still have access to your modded blocks. If you do, fire up your previous world (that you should have backed up) and you should be good.
If things go awry, you can downgrade back to 1.12 and use your backed up world.