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I planted Shasta daisy seeds a few weeks ago indoors, on a window sill. The plants are now a floppy four inches tall, transferred to 4 inch pots, and still indoors.
My average last frost date is April 21 or so, here in western Washington. I'd like to put them outside so I can start more seeds on the sill. Is it okay to do that while it is yet the beginning of March? Will they be okay if I try hardening them off, bringing them in for the nights for a while but putting them outside during the day

Yes, getting floppy probably means there isn't enough light for them on the window sill, so getting them outside would be a good idea.

Shasta daisies are cold hardy perennials, so there is no reason to wait for frost free date. Having been indoors though, they will have to be hardened off, get used to the outdoors gradually. So yes, start putting them outdoors, and bringing in for a little while. Start with putting them outdoors in a spot that doesn't get direct sun and then each day move them to areas that get more sun. As well as the temperature, indoor plants have to get used to how much light there is outside.

Thank you! I'm working on hardening them off.