Do I have to buy upgrade materials from the store to reach level 5?
Solution 1:
To fully upgrade them on your first play through, yes. However, after you beat the game a New Game+ unlocks which carries over all your upgrades and weapons from the previous game. This makes buying from the shop unnecessary if you plan on doing a second play through.
However, unless you plan to fully upgrade every weapon in the game on your first play through, you won't need to concern yourself with money. You'll have enough money to clear out the lost & found, in addition to fully upgrading at least two weapons, as well as partially upgrade a few others.