user12824: En passant is somewhat of a rare move in chess. Checkmating by en passant, however, is an even rarer occurrence. But suppose both sides had the same goal to help cause a checkmate to their opponent via en passant.
What is the minimum amount of moves needed by both White and Black to give checkmate by en passant?

user15543: There are four kinds of en passant checkmates.

The pawn moves, checking the king itself.
The pawn moves, checking the king, while also allowing another piece to check as well. This is a double check.
The pawn moves, not checking the king, as to allow another piece to check the king. This is a discovered check.
The pawn moves, not checking the king itself, as to allow two other pieces to check the king. This is a double discovered check.

Here are the fastest possible games for each category.
Reversing each game, adding on a ply, provides for Black's fastest mates. These are sourced from the same Lichess forum linked in the accepted answer.
[Title "Type #1"]
[FEN ""]

1. e4 f5 2. exf5 Kf7 3. Qg4 h6 4. b3 Qe8 5. Bb2 g5 6. fxg6#


[Title "Type #2"]
[FEN ""]

1. e4 d5 2. d4 dxe4 3. d5 Kd7 4. Bc4 Nc6 5. Bg5 Qe8 6. h3 e5 7. dxe6#


[Title "Type #3"]
[FEN ""]

1. e4 e5 2. g4 Nc6 3. g5 d6 4. Qh5 Kd7 5. Bh3+ f5 6. gxf6#


[Title "Type #4"]
[FEN ""]

1. c3 f5 2. h4 Kf7 3. Qb3+ Kg6 4. Qf7+ Kh6 5. h5 a6 6. d4+ g5 7. hxg6#


user8201: Thanks for the subject! A natural, fun extension of the question is to ask what is the shortest Unique Proof Game ending in e.p. checkmate. The idea with one of these is you are given just the end state diagram, and the total number of moves, and you must figure out the unique game leading to that point.
I think the current record holder is the following:

[title "Gerd Wilts & Norbert Geissler - RML - 05/1996"]
[fen ""]
[startply "14"]  

1. f4 e5 2. Kf2 Qh4+ 3. Kf3 Qf2+ 4. Kg4 h5+ 5. Kh3 h4 6. e4 d5+ 7. g4 hxg3ep# *

Position after Black's 7th move.  
I honestly don't know if this is the best possible: maybe there is another position with a shorter unique proof game.

user15415: The same question has also been discussed and answered in the lichess forum, and as mentioned by others already, the answer is 11 half-moves for white and 12 half-moves for black. You might find some interesting examples and additional info there. Not knowing that this problem had already been solved before, I did a brute-force proof back then (see my post in the aforementioned thread), since that is feasible for 10-12 ply, i.e., disproving the existence of a solution up to 10 ply by an exhaustive search and finding solutions with 11 and 12 ply, respectively.
Here are displays for said games.
[FEN ""]

1. e4 e6 2. e5 g5 3. Nc3 Nh6 4. Qh5 Ke7 5. Ne4 f5 6. exf6#


[FEN ""]

1. f3 e5 2. Kf2 Qf6 3. Kg3 h5 4. h3 h4+ 5. Kh2 Qxf3 6. g4 hxg3#


user8860: According to
this page
the shortest known mate by en passant capture is in 5.5 moves
(that is, White mates on move 6), and was published by Benko in
Chess Live & Review in 1976:
[Title "Pal Benko's 5.5 move e.p. helpmate"]
[Fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w - - 0 0"] 

1. e4 e5 2. Qh5 Nc6 3. g4 d6 4. g5 Kd7 5. Bh3+ f5 6. gxf6#


12824: <p>En passant is somewhat of a rare move in chess. Checkmating by en passant, however, is an even rarer occurrence. But suppose both sides had the same goal to help cause a checkmate to their opponent via en passant.</p>
<p>What is the minimum amount of moves needed by both White and Black to give checkmate by en passant?</p>
 15543: <p>There are four kinds of en passant checkmates.</p>
<ol>
<li>The pawn moves, checking the king itself.</li>
<li>The pawn moves, checking the king, while also allowing another piece to check as well. This is a double check.</li>
<li>The pawn moves, not checking the king, as to allow another piece to check the king. This is a discovered check.</li>
<li>The pawn moves, not checking the king itself, as to allow two other pieces to check the king. This is a double discovered check.</li>
</ol>
<p>Here are the fastest possible games for each category.
Reversing each game, adding on a ply, provides for Black's fastest mates. These are sourced from the same Lichess forum linked in the accepted answer.</p>
<pre><code>[Title &quot;Type #1&quot;]
[FEN &quot;&quot;]

1. e4 f5 2. exf5 Kf7 3. Qg4 h6 4. b3 Qe8 5. Bb2 g5 6. fxg6#
</code></pre>
<hr />
<pre><code>[Title &quot;Type #2&quot;]
[FEN &quot;&quot;]

1. e4 d5 2. d4 dxe4 3. d5 Kd7 4. Bc4 Nc6 5. Bg5 Qe8 6. h3 e5 7. dxe6#
</code></pre>
<hr />
<pre><code>[Title &quot;Type #3&quot;]
[FEN &quot;&quot;]

1. e4 e5 2. g4 Nc6 3. g5 d6 4. Qh5 Kd7 5. Bh3+ f5 6. gxf6#
</code></pre>
<hr />
<pre><code>[Title &quot;Type #4&quot;]
[FEN &quot;&quot;]

1. c3 f5 2. h4 Kf7 3. Qb3+ Kg6 4. Qf7+ Kh6 5. h5 a6 6. d4+ g5 7. hxg6#
</code></pre>
 8201: <p>Thanks for the subject! A natural, fun extension of the question is to ask what is the shortest <strong>Unique Proof Game</strong> ending in e.p. checkmate. The idea with one of these is you are given just the end state diagram, and the total number of moves, and you must figure out the unique game leading to that point.</p>

<p>I think the current record holder is the following:</p>

<pre><code>
[title "Gerd Wilts & Norbert Geissler - RML - 05/1996"]
[fen ""]
[startply "14"]  

1. f4 e5 2. Kf2 Qh4+ 3. Kf3 Qf2+ 4. Kg4 h5+ 5. Kh3 h4 6. e4 d5+ 7. g4 hxg3ep# *
</code></pre>   

<p>Position after Black's 7th move.  </p>

<p>I honestly don't know if this is the best possible: maybe there is another position with a shorter unique proof game.</p>
 15415: <p>The same question has also been <a href="https://lichess.org/forum/general-chess-discussion/competition-create-the-shortest-game-with-ep-and-mate-pxp-ep-?page=1" rel="noreferrer">discussed and answered in the lichess forum</a>, and as mentioned by others already, the answer is <strong>11 half-moves for white</strong> and <strong>12 half-moves for black</strong>. You might find some interesting examples and additional info there. Not knowing that this problem had already been solved before, I did a brute-force proof back then (see <a href="https://lichess.org/forum/general-chess-discussion/competition-create-the-shortest-game-with-ep-and-mate-pxp-ep-?page=2#13" rel="noreferrer">my post in the aforementioned thread</a>), since that is feasible for 10-12 ply, i.e., disproving the existence of a solution up to 10 ply by an exhaustive search and finding solutions with 11 and 12 ply, respectively.</p>

<p>Here are displays for said games.</p>

<pre><code>[FEN ""]

1. e4 e6 2. e5 g5 3. Nc3 Nh6 4. Qh5 Ke7 5. Ne4 f5 6. exf6#
</code></pre>

<hr>

<pre><code>[FEN ""]

1. f3 e5 2. Kf2 Qf6 3. Kg3 h5 4. h3 h4+ 5. Kh2 Qxf3 6. g4 hxg3#
</code></pre>
 8860: <p>According to
<a href="https://madchess.wordpress.com/2012/06/17/shortest-legal-en-passant-mates/" rel="noreferrer">this page</a>
the shortest known mate by <em>en passant</em> capture is in 5.5 moves
(that is, White mates on move 6), and was published by Benko in
<em>Chess Live &amp; Review</em> in 1976:</p>

<pre><code>[Title "Pal Benko's 5.5 move e.p. helpmate"]
[Fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w - - 0 0"] 

1. e4 e5 2. Qh5 Nc6 3. g4 d6 4. g5 Kd7 5. Bh3+ f5 6. gxf6#
</code></pre>
