Fama and French calculate the factors based on CRSP data. The fastest that CRSP data is updated is monthly which is why Ken French updates the data on his website at monthly intervals.
If you need daily updates, you would have to construct the 2x3 sort by ME and BE/ME yourself. French's website publishes the breakpoints (based on all NYSE stocks) for the portfolios. But as far as I know, he does not publish what the portfolio constituents are. For the SMB and HML factors, the portfolio sorts are formed at the end of June each year and held constant throughout the year. You can use CRSP and Compustat data (or any other source that gives you the market cap and book equity) to figure out which stock is member of which portfolio.
If you have access to the returns of all stocks listed on NYSE, AMEX, and NASDAQ at the end of each day, you can then construct the factor returns.
P.S.: There may be data vendors that sell the factor returns at a daily frequency, but I am unaware of them.