Will probably need to be back from the beach a ways to do well. Most stock sold now is on dwarfing roots, which makes them less sturdy at the bottom than old standard trees. Fully dwarfed specimens may need permanent support. Planting now you would be buying a specimen growing in a pot, it would be possible to do this if someone can water it during the rest of the summer. Diseases could be dealt with acceptably by selecting a resistant cultivar, see Washington State University Cooperative Extension Service publications on fruit trees for recommendations. Note that with apples you usually need two different yet compatible kinds for full cropping. If deer have access to the site these are liable to browse the tree(s), also bears are attracted to the fruits.

Anything that can be kept watered can be planted this time of year. If it's on the beach, a shore pine might be more appropriate, Pinus contorta. An apple would not do well on the beach as stated above. Or how about a Pacific crabapple? they do well on sandy soil near the beach. Malus fusca. They do get as big as a small apple tree but the fruit's not sweet.

If it can't be kept watered then you're best off waiting til fall, like late September or October, to plant.

What a wonderful way to commemorate a strong and generous member of the family. I love your idea!

I did something along somewhat similar lines to mark the wedding of my sister into the family name of Banks by planting a rosa banksiae ("Lady Bank's" rose). If apple trees won't do well in your intended site (keep in mind it might not get too much in the way of salt spray at the family cabin, depending on where it is situated on Puget Sound, so don't automatically be discouraged from planting an apple tree), perhaps you can find something else to symbolize Nana. A plant with her name or certain attributes would be nice. Also, from my decidedly brief web research on the topic, I understand that rugosa roses do well in coastal sites. Some are fragrant (her "sweetness" coming through as a scent rather than a taste) and "rugosa" shares Latin roots with "rugged" (not a "foundation", but at least showing strength and resilience). Further, certain blueberries are salt water tolerant and are among the longest lived fruit-bearing shrubs, so the sweetness would be shared by family members long into the future. A nice legacy.

The very best of luck to you with your project.

Rain