A quick approval process is provided by some insurance companies for online applications.
Some carriers will process an online application within a few minutes if all health questions are answered "no".
Any "yes" answers to health questions will always delay an application no matter where you apply or to which insurance company because it will have to be reviewed by a person, an underwriter.
It does not matter where you find an application online, you are always buying direct from the individual carrier, not from the agency. Agencies are only helping you apply. Carriers completely control their own products, so agencies make no difference to plan design, price, or how fast the carrier processes an application.
If all your health questions have a "no" answer, i.e., you have no pre-existing conditions whatever, you can virtually be assured of accceptance, no matter how long it takes the carrier to process the application.
If medical records are needed, the application will be delayed even more. Depending on the carrier, either the carrier requests the records or you will have to have them sent in. Either way, that will not be a fast process, no matter which website you apply at.
The only health insurance that is always processed immediately with a reply by return email is short term coverage (most carriers). Those applications are short and simple and policies can be issued electronically without prior human review.