Something like a flip flop
Solution 1: A latching relay.*
Have a look at latching relays.

Figure 1. There's an animation of one on [HomoFaciens.de]
(https://www.homofaciens.de/technics-base-circuits-relay_en.htm) that explains the operation very well. (Click the &quot;animate&quot; link in the article to see it in action.)
Solution 2. A relay flip-flop.



simulate this circuit &ndash; Schematic created using CircuitLab
Figure 2. A weird relay toggle circuit.
How it works:

On power-up both relays are off. The chuck solenoid, SOLN, is de-energised. (This is good as it gives you a predictable power-up state.)
When FOOTSW is pressed RLY2 is powered on. SOLN will then be powered on.
While FOOTSW is pressed RLY1 is shorted out and cannot energise.
When FOOTSW is released RLY1 is connected in series with RLY2 and will energise. Both relays now receive half of the supply voltage. (See Note 1 below.)
When RLY1 energises contacts RLY1a and RLY1b switch over setting it up for the next FOOTSW operation.
On the next press of FOOTSW the bottom of RLY1 is grounded (giving full voltage to RLY1 as long as FOOTSW is held) and RLY2 is shorted out. It will switch off.
When FOOTSW is released the circuit resets to the original state.

Note 1:
The circuit relies on the relays energising in series and on their own. That means operating on half of supply voltage and full supply voltage. You'll need to rummage through your selection of relays and find a pair that will operate when the coils are connected in series.
Figure 3. Timing diagram.
             ____                ____              ____
FOOTSW  ____|    |______________|    |____________|    |_______
                  ______________                        _______
RLY1    _________|              |______________________|
             ___________________                   ____________
RLY2    ____|                   |_________________|


From the comments:

... but I need a pulse in different directions for the duration of holding the pedal.

Figure 4. Updated timing diagram.
             ____                ____              ____
FOOTSW  ____|    |______________|    |____________|    |_______
                                 ____                            
SOLN-O  ________________________|    |_________________________
             ____                                  ____
SOLN-C  ____|    |________________________________|    |_______


So the latching would chose the direction and second contacts on pedal would just activate it.

Correct. See Figure 5.



simulate this circuit
Figure 5. Schematic updated for timing diagram of Figure 4.