I know,’No sooner had I reached the station than the train leave’ is a common usage.

However, the following sentence should be also acceptable.

No sooner had I reached the station did the train leave.

Then, how is this sentence formed?

In order for this sentence to be accepted, the inversion, ‘did the train leave’ should take place by fronting the phrase, ‘no sooner ...’.

→ The train left [no sooner had I reached the satiation].

But I don’t think this sentence is correct because it doesn’t have a conjunction. Or I wonder ‘no sooner’ somehow works as a kind of conjunction...

"No sooner had I reached the station did the train leave." Is a hundred percent acceptable as well.

This answer is a bit complex, so bare with me here.

When it comes to building sentences, there are two different types of "clauses that can be used.

Dependent
Example: "When Sally went to the store."

Independent:
Example: "She bought 28 apples."

One can stand alone because it's a complete thought, (the independent clause) while the other cannot, (The dependent clause).

Now, when it comes to building sentences, we can use any combination of these clauses as long as it doesn't run-on* (more about this later).

A fragment (not a complete sentence, but I figured that I'd add this in here, because even native speakers can struggle with avoiding these) is made up of one dependent clause.

A simple sentence is made up of one Independent Clause.

A complex sentence is made up of one dependent clause and one independent clause.

Finally, a compound is made up of two independent clauses.

Only the Compound sentences require a conjunction.

The sentence you're asking about falls in the "Simple sentence" category, therefore it doesn't need a conjunction.

The sentence is fairly long for a simple sentence, yes; however it still only has one idea (even if it is a detailed one).

Now, your tensing is a little odd, however it gets the point across (the listener will most likely understand what you meant).

*A run-on sentence is basically a sentence that is too long. For example:

"I went to the store, and Susie played video games, and Tyler went skiing, and Fiona ate her lunch."

To be fair this is a fairly extreme example; however, it can be very common among children.

Hope this helps! If anything's confusing, please reach out. I tried to keep a balance between too much information, and not enough.