Definitely, there will an inventory. Haven't thought about whether to change the axe eventually.
Well there are a lot of options you can do even if you want him to keep the same axe. It really depends how complex you're willing to get with it all. If you want him to keep the same axe then the passive stats of the weapon could always be improved by Lanya at the forge, without having to give him a whole new model for a weapon. Maybe she could do this once or twice, until she can't physically upgrade the weapon anymore. But if you persist and ask if there is anything else she can do, she could say that she heard of a rumor about a type of mystical stone from deep within the mountains, that she could use to turn into a unique type of sharpening stone. If you find it, then once a week she could sharpen your weapon for you and you could get a small damage boost or critical strike boost. Of course she will require a small fee. And If you plan on having buildings upgradeable, then you could upgrade her forge. As a result she could get access to better blacksmithing equipment, and make even better sharpening stones. Now instead of giving you one bonus stat over another, she is able to combine the damage and crit boost together.
If you're not married to that axe and are open to the MC wielding other kinds of weapons like swords or maces, then she could always build those as well. But you have to go out and find some blacksmithing plans. You do this by going out and raiding monster caves, ancient ruins, or bandit encampments then take the plans you find to her. After you bring her back the plans, she would (unsurprisingly) claim that she needs other certain materials to actually craft the weapon you want. If you want something better than a basic two-handed sword, axe, or mace that is. Then depending on how good you want the quality of the weapon to be, that could lead to other side quests where you have to hunt for rarer materials the same way you did for the blacksmithing plans. Of course the rarer the materials are, the harder they will be to obtain.
Then there is always magical enchantments. Perhaps this is where Verna can come into play. Same as with the blacksmithing plans or rare materials, she could require you to find magical enchantments on ancient scrolls lost out in the world. She will give you some leads as to where to go, but she isn't sure what you will find exactly. So you go out, kill some monsters or other bad guys, loot their corpses and find an enchantment. You bring it back to her and she can now enchant your weapon to do bleed damage or possibly ignore some armor. That type of thing.
These are just some examples. I hope they helped pique your interest. And if this all sounds way too complicated or like a programing nightmare, then forget I said anything.