Organic vegetable online stores fall into three categories:

I) Organic Farms that grow vegetables in their farms and distribute

1. OrganicSiri.com is the only organic store who grow everything they deliver as far as I know. They have farms in different parts of Telangana and deliver across Hyderabad based on online orders direct from farms, without a physical store. They are very reliable and reputed.
2. NaturePinks farms near Hyderabad produce organic vegetables and fruits and distribute on OrganicSiri.com. Quite an old operation in organic food space.

This type of stores are most trustworthy but also very rare. Since they grow all their organic food themselves, there is perfect traceability of where the produce came from and you are always able to check back with the farmer directly for any questions. Their limitation is that the variety of produce is somewhat limited to what they grow. For example, they cannot offer cool season vegetables like green peas in summer season since they can grow them in Hyderabad only in winter. They will never have apples unless they also have a farm in the North. Also, their stocks are also limited in availability and cannot scale beyond what they have planned and produced.

II) Organic sellers who partly grow and partly procure from other farmers

1. Farm2Fridge is a seller who seems to grow a part of what they sell and procure the rest. They have been there for a while and operate in basket delivery model.
2. Sage Organics seems to have a small farm and also procure from other farmers. Their site does not have an option to order online, it is not clear if they are operational in retail.

This type of stores grow some of their produce and procure the rest from other farmers across the country. The benefit is that they can obtain a large variety of produce round the year. Even stuff like apples, capsicums, potatoes will be procured from the cooler parts of the country any time. The disadvantage is that procurement means reduction in reliability of how organic the products really are. Since it is impossible for the seller to monitor farmer's practices, significant chance of mixing of non organic produce exists. These stores also are seen to work with middlemen for supplies. Over time, it appears that these sellers depend more on procurement and produce less since production is more difficult of the two.

III) Organic sellers who aggregate and sell in physical shops or online

1. Daman Organic Living has a couple of large format stores and seem to procure variety of products. They seem to operate in franchise model and are mostly walk in stores, though it appears they may be selling online also now.
2. Green Station is a venture of Terragreens and operates on franchise model. They seem to have two physical stores with a widely spread procurement and also are operating online.
3. Gofarmz seems a group of software engineers who built the site to procure and distribute organic produce. Interestingly they have listed some known aggregators as the farmers they procure from; this is probably the way most of the online and offline stores are forced to source, given the current limitations of organic farm produce availability.
4. Bigbasket also has an organic brand that they sell on. I believe they procure from agents in multiple states and it is interesting that sometimes their organic variants are cheaper than their non-organic variants (not sure how it is possible). They deliver in Hyderabad though feedback seems mixed on quality and delivery.

These stores are complete marketplaces who procure their produce through middlemen or by liaising directly with farmers. Some of them try putting in some quality checks on the production process to ensure the produce is organic. They also have the benefit of a wide range of products round the year, since they source from multiple locations. The disadvantage is that with the scale of operations and spread of procurement, it is never easy to enforce purity. Further most of them operate in franchise model and need to have good controls to confirm the produce is truly organic.

I will keep updating the stores and links periodically. It is interesting to note that many new stores open up each year and most of them close down due to poor organic produce availability or bad economics.

It is however commendable to see the work the stores put in to make organic food accessible to everyone, even though some of them have better reliability than others.

[Please upvote and comment so I can add further updates]