If i use the lean model and test my MVP with limited number of customers (suppliers) available for my user (buyer) to choose from wouldn't it result in creating a bad user experience as they will not find what they are looking for?
So long as the suppliers are able to deliver a basic level of service that would create a positive experience for the purchaser, I'd say that it's reasonable to conduct the experiment. Marketplaces always face liquidity issues in the early days.
Happy to talk to you in more detail in a call to better understand your current MVP and roll-out plans and provide some more specific advice to you.
You should wait! You need to do market testing with real buyers that would have a real application for your site. If you roll out prematurely and do not provide a positive experience, it will be costly.
Offer your service for free for a limited number of potential buyers in exchange for their feedback. They become your beta testers. It does not have to be free forever. Let them know at some point the beta project might be over.
I am offering free calls to first time callers. Use this link to schedule and we'll talk more in detail about doing this the right way.
It depends whether this is your first market or not. Going live with limited supply is usually inevitable when it's your initial marketplace launch. In this case, you should look for a differentiating benefit that you can control and deliver to your demand side customers regardless of how quickly your supply ramps up.
For example, when Lyft launched, there weren't many drivers (supply) so a taxi might be faster, but if you waited they would drive you wherever you wanted, even out of the city, whereas a cab usually wouldn't. Getaround offered lower than Zipcar prices in exchange for sparse supply and longer wait-times to get car owners to approve your rental request.
But if you have already launched in one market, I would focus on that before launching another. If you figure out those dynamics first, you will waste less time and money in the new market.