how to split an image with a grid and export automatically every frame of the grid in a single file in photoshop or illustrator?

Select the slice tool (under the crop category), with the slice tool select your entire image.

Right click on your selection and choose ‘Divide Slice’, enter the horizontal and vertical dimensions of your choice.

Now choose ‘Save for Web’, select all your slices and choose your format and compression. Click save and make sure it says ‘All Slices’ under ‘Slices’.

You could record all this in an action to re-use it.

Edit: Forgot to mention this is for Photoshop.