yes, you could add a timer module that are available everywhere. But since they work with low DC voltages, you would need to also use an additional external relay that would switch the 220 V pump on and off. The timer module would switch the control input of the external relay, and the timer module would be triggered by the relay output of your VFD as already mentioned above.
Usually I would say a module based on a programmable interval timer like the ubiquitous NE 555 would be totally sufficient, but I have learned that nowadays, you get a microcontroller-based module at the same price . So you could buy a TRM01 timer module (a timer module based on the microcontroller STC 8G1K08, which is a Intel MCS-51 (8051) specimen) and is programmed to allow up to 120 s delay. I would choose the 24 V version as this voltage is the most common for accessories in a VFD control cabinet. There was a thread about exactly this → here the other day. It was for a Huanyang VFD, so the relay pins have other labels than your Hitachi S1, but you see how it works. You use the normally open (NO) contact of the VFD’s relay output and connect it to (e.g. the VFD’s internal) 24 V, and you connect the common contact to trigger the X1 input of the TRM01 module. The advantage of this module is that it already contains the external relay that can switch 250 V AC 10 A! So you could connect your pump directly to it. The module can have four modes of operation, of which one is to insert a delay before shutting the load (your pump) off.
Note that on the thread mentioned, the forum user wanted to use a solid state relay (SSR), here the ubiquitous FOTEK SSR-40 as described by buildbotics on this page, but there are some disadvantages in using a solid state relay compared to a “true” electromagnetic relay. The advantage of SSrs are that they need only an extremely small control current to be switched on or off. But their disadvantages are that they are made of only semiconductors, so on overload they can be destroyed or start malfunctioning easily without you noticing it, and the other disadvantage is that to hold their state, they need a relatively high switched current on their load side, of at least approx. 100 mA, which means if the load you switch draws less current than this, you cannot be sure that the switching “on” state of the relay will persist, or the relay reverts its state back. An electromagnetic relay can switch much lower load currents reliably, and it is either functional or broken. And usually they have a transparent housing where you can see the physical state of the contacts (and hear them, they “click” when switched). SSRs don’t “click”.
But the real problem with the SSR described, the FOTEK SSR-40, is that
you only find counterfeit FOTEKs on the market. I did not find an original.
Anyway if you choose the TRM01 module, it already has a “true” electromagnetic relay onboard (the blue box on the pcb) that will switch a load of max. 250 V AC / 10 A. Just be sure to select the version for 24 V for the control circuit and the microcontroller supply.
Otherwise, when using just a NE555-based timer, when selecting an external relay, be it a solid state or an electromagnetic relay, you need one that is rated on its control input by preferably 24 V DC and that can switch a load of at least 220 V AC / 0.5 A (which is your pump).
If one day you want to also switch your dust collection system or workshop vacuum cleaner, it could be that 10 A are not enough. In this case you would need an additional relay with that capability, but with the TRM01 module, you already can switch 250 V AC / 10 A with the blue onboard relay.