Selling Online Before Taking Possession of Items
Question

The concept is that when a customer makes a purchase from the online-store, he pays using a payment processor with the option of either card or bank. The payment processor receives the money and takes a fee like 2% for the transaction, then the system (My Website) will automatically send that order to the distributor to process the order. There are two options in dealing with the the distributor: 
1) Is to have money pre-deposited at the distributor's bank account so that when a purchase is made they just pack and send the product to the customer. 
2) Is for them to ship the product to the customer upon order completion, and then allow me to pay by invoice later on. Shipping takes 1-3 days approximately. Then later on I will have to make a withdrawal from the payment merchant to get the money that the customer paid. Another option when it comes to the customer paying is to allow the choice of invoice in the payment processor instead of just card and bank payments. 
However, invoice creates a contract between the customer and the payment merchant so that the customer can pay within 14 days or by installments, but in installments Riba will be between those two parties depending on the contract. The payment merchant will give me the money in full even though the customer has not completed the payment to them. In all options the products are sent directly to the customer from the distributor upon purchase from my online-store. Is this kind of business allowed?
Praise be to Allah.According to what we understand from the question, there are four parties involved in this transaction:

The purchaser
Your website
The distributor
The payment processor.

Can you pay via payment processor in return for a percentage per transaction?
It is permissible for the purchaser to pay the price via the payment processor, in return for 2% of each transaction. This is acting as a proxy in return for a fee, and there is nothing wrong with it.
Can you sell an item before taking possession of it?
It is not permissible to sell an item before taking possession of it , and moving it from the distributor’s place, because of the report narrated by An-Nasa’i (4613), Abu Dawud (3503) and At-Tirmidhi (1232) from Hakim ibn Hizam (may Allah be pleased with him), who said: I asked the Prophet (blessings and peace of Allah be upon him): O Messenger of Allah, a man may come to me wanting to buy something that I do not possess, so I sell it to him then I buy it for him from the marketplace. He said: “Do not sell that which you do not possess.” The Hadith was classed as authentic by Al-Albani in Saheeh An-Nasa’i.
Ad-Daraqutni and Abu Dawud (3499) narrated from Zayd ibn Thabit (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) forbade selling an item where it was bought , until the merchants move it to their own places. The Hadith was classed as sound by Al-Albani in Sahih Abu Dawud.
How to fix the issue of selling items before possessing them
Ways of resolving this problem are as follows:

Just promise the customer that you will buy the item, then take possession of it, and then sell it to him. When you have bought it and taken possession of it, then do the transaction and send it to him.
It is permissible to be an agent acting on behalf of the distributor, selling the item for him in return for commission, or to be an agent acting on behalf of the purchaser, buying the item for him at the usual price, and agreeing with him on a fee in return for doing that.
There is also a third permissible way, which is the Salam transaction. This means selling an item to the customer that is clearly described and defined, in such a way that there can be no cause for dispute, which you commit to delivering to him at a specified time, on condition that you receive the money from him when drawing up the contract with him, even if that is by him depositing it in your account, because that is the same as receiving it in your hand, in terms of the ruling. It is not right for the money to remain with the payment processor.

Thus you will know that the problem is when you do not have the item in your possession before selling it to the customer.
This problem may be solved in one of the ways mentioned above, but the first way is not appropriate in your case, because you do not take possession of the item from the distributor before selling it.
Can you act as a proxy in return for a fee?
Acting as a proxy in return for a fee is valid in your case, if the distributor agrees to that and regards you as his agent, in return for commission that he pays to you. In that case, it is not valid for you to deposit money in his account beforehand, because you are his agent and acting on his behalf; you are not buying from him.
However, the second way you mention is valid, which is your withdrawing the purchaser’s money from the payment processor and sending it to the distributor, in which case you are acting as an agent, either for the seller, by selling and receiving the price of his item, or for the purchaser, by buying for him and paying on his behalf. Nonetheless, this is on condition that there should be an agreement between you and the other party who will appoint you as his agent, and that you should state your fee for acting as his agent.
Salam transaction
As for the Salam transaction, it is not valid in your case, because the condition for it being permissible is that you should take the price in full at the time of drawing up the contract, and the money should not remain with the payment processor.
In conclusion, so long as you do not have possession of the item and you cannot take possession of it before selling it, or obtain its price in full from the purchaser, then you can only act as an agent for the distributor.
Can the customer pay installments through the payment processor? 
What you have mentioned about the customer paying installments through the payment processor is not permissible, and it comes under the heading of Riba as you mentioned, because if the one who provides the payment pays the price on behalf of the purchaser, then asks the purchaser to pay him more, then that is Riba.
What is permissible is for the customer to give the money to him (the agent), so that he can send it to the seller, in return for commission; in that case he will be acting as an agent or proxy in return for a fee, as mentioned above.
For more details, please read the following answer: 334744
And Allah knows best.