Replacing periods with commas

This general stylistic device is called asyndeton:

'A rhetorical figure which omits the conjunction'
^{Oxford English Dictionay}

'omission of the conjunctions that ordinarily join coordinate words or clauses
(as in "I came, I saw, I conquered")'
^{Merriam-Webster}

See e.g. here for more examples. There doesn't seem to be a specific name for the case when what is being joined are main clauses as opposed to subclausal groups of words.

One of the most frequently quoted examples of this literary scheme is Winston Churchill's address 'We shall fight on the beaches':

We shall go on to the end, we shall fight in France, we shall fight on the seas and oceans, we shall fight with growing confidence and growing strength in the air, we shall defend our Island, whatever the cost may be, we shall fight on the beaches, we shall fight on the landing grounds, we shall fight in the fields and in the streets, we shall fight in the hills; we shall never surrender…

Another famous example, where the entities being joined by the commas aren't main clauses, is this part of Lincoln's Gettysburg address:

…and that government of the people, by the people, for the people shall not perish from the earth

The grammatical term for all this is asyndetic combination (CGEL, pp. 1276, 1741-1742). In the specific case of your example and Churchill's address, it is asyndetic combination of main clauses.