Should there by a comma in the sentence:
On that day in late September I went to look at puppies.
Should there by a comma in the sentence:
On that day in late September I went to look at puppies.
Should there by a comma in the sentence: "On that day in late September I went to look at puppies."?
Yes, and no, or maybe two, it depends entirely upon the context.
In each case a different part of the sentence is stressed as important.
In general, think of what the sentence means, of what is most important, then slowly say it out loud. You will naturally pause at certain places and not at others. Those pauses are where the commas belong.
Introductory clauses normally have a comma after them. The use of a comma indicates that everything that comes before is a dependent clause, while everything that comes after is an independent clause.
As mentioned in a comment, this is what the Purdue Online Writing Lab says:
2. Use commas after introductory a) clauses, b) phrases, or c) words that come before the main clause. …
b. Common introductory phrases that should be followed by a comma include participial and infinitive phrases, absolute phrases, nonessential appositive phrases, and long prepositional phrases (over four words).
Having finished the test, he left the room.
To get a seat, you'd better come early.
After the test but before lunch, I went jogging.
The sun radiating intense heat, we sought shelter in the cafe.
Similarly, The Chicago Manual of Style (17th ed.), 6.24, says this:
When a dependent clause precedes the main, independent clause, it should be followed by a comma. A dependent clause is generally introduced by a subordinating conjunction such as if, because, or when …
If you accept our conditions, we shall agree to the proposal.
Until we have seen the light, we cannot guarantee a safe exit from the tunnel.
Whether you agree with her or not, she has a point.
In the example sentence in the question, on that day in late September is an introductory and prepositional phrase, as well as a dependent clause. By the majority of style guides and common conventions, a comma would be used:
On that day in late September[,] (dependent clause) I went to look at puppies. (independent clause)
While it would still be meaningful without a comma, it could possibly lead to some confusion.
For instance, without the comma, the entire sentence as it's written could be considered a single clause. But if that's the case, then it's also incomplete:
On that day in late September
[when/that]I went to look at puppies— (everything is a single dependent clause)
What? What happened on that day in late September you went to look at puppies?
Note, too, that this is also, so to speak, moving the goalpost of the comma. If you follow the convention of using a comma at the end of a dependent clause, the comma here has just been moved from after September to the end of the phrase as a whole, and the independent clause itself has simply been omitted.
For instance, the following is a possible continuation of the sentence:
On that day in late September I went to look at puppies[,] I was run over by a car as I crossed the street.
Without the comma, it's still possible to parse the sentence as having a dependent clause and an independent clause. So, it's not wrong per se.
You don't have to have a comma in order for there to be such a division, and you don't have to parse the comma-less version as a single dependent clause with something missing.
However, if you want to convey unambiguous meaning to every reader, it's probably better to include the comma after the dependent clause—just to make sure that it's not misinterpreted.