Don't think so. The request overwrites the data pull each time.
One workaround is to calculate the start and end date for the report - end now, start now minus 1 day, for example (careful with your date format). That way you get a new sheet each day. You can then use Query function to merge the sheets together into one single sheet.
A warning: each new report sheet will have columns A-Z, and if you are running many rows of data you will hit the cell limit per sheet fairly quickly. If reporting each day, consider making one sheet for each calendar month. You can then combine the results into one master sheet per year, to avoid those limits.