Purée the grapes in a blender or food processor, in batches if necessary, until very smooth. Pour the purée into a fine sieve and press hard with a spatula to extract as much juice as possible. Purée the grapes again and put them back into the sieve. Press again and reserve the juice for another recipe. There should be about 2 cups of grape purée.
Pour the purée into a small saucepan with the sugar and lemon juice and bring to a boil. Simmer, stirring often and scraping the sides, until the purée turns glossy and darkens slightly, about 30 to 45 minutes.
Preheat the oven to 200 F using convection mode if available. Line a large-rimmed baking sheet with a silicon baking sheet. Pour the purée onto the mat and spread into an 1/8-inch thickness using an offset spatula, leaving a 1-inch border of space around the edge. Bake 1 1/2 to 3 hours, rotating the tray halfway through, until the purée is tacky and not quite dried in the center. Let stand at room temperature until fruit leather peels easily from the tray, 24 hours to 48 hours.
Place a piece of parchment paper over the fruit leather and invert. Peel off the silicon baking sheet and starting at the long end, roll up the fruit leather and parchment. Using scissors or a sharp knife, cut the fruit leather roll into 1-inch segments. Store in a resealable plastic bag for up to 1 week at room