Praise be to Allah.
If a person has a share of a shop, it is permissible for him to rent it out for a fixed, known amount, to his partner or to someone else, according to the basic principle that it is permissible to rent out houses, shops and other things which may be used and benefitted from and still remain. Renting something that is shared with a partner is permissible according to scholarly consensus.
Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni (5/317):
Section: What it is permissible to rent or hire out. It is permissible to rent or hire out anything that can be used in permissible ways but still remains, such as land, houses, slaves, animals, clothing, pavilions, ropes, tents, bags, lamps, reins, swords, spears and so on.
In al-Mawsoo‘ah al-Fiqhiyyah (1/263) it says: Renting or hiring out items of which ownership is shared with others: if the item in question is shared and one of the two partners wants to rent out the benefit of his share of it, then it is permissible for him to rent it out to his partner, according to scholarly consensus. With regard to renting out to someone other than one’s partner, then according to the majority of Hanafis, Shaafa‘is and Maalikis, and according to one report narrated by Ahmad, this is also permissible, because renting or hiring is one kind of transaction, so it is permissible to rent out shared property just as it is permissible to sell it, and shared property may be benefitted from by turns [i.e., each partner may benefit from it for a certain period]. Hence it is permissible to sell it. It says in al-Mughni: Abu Hafs al-‘Akbari favoured the view that it is permissible to rent out shared property to someone other than one’s partner. Ahmad suggested that, because it is in his possession and he could rent it out to his partner, therefore he could rent it out to others, like selling. And because it is permissible if both partners do that together, so it is permissible for one of them to do that with his share on his own, as is also the case with selling. End quote.
Conclusion: it is permissible to rent out your share of the shop on its own, or with the fixtures and fittings that are in it, such as equipment, shelves and so on.
With regard to your share of the goods – if applicable – it is not acceptable for you to rent them out, because renting is a contract that has to do with benefitting, and goods can only be benefited from by selling them; rather the goods are to be evaluated and sold to the renter, if he wants that, even if it is by instalments.
It is permissible to delegate him to sell them, for free or in return for payment, according to whatever you agree, or you can take your goods and sell them to whomever you want.
And Allah knows best.