You can request a copy of a deceased person's application for Social Security here:
https://www.ssa.gov/forms/ssa-711.pdfbut it won't tell you anything you don't know, except where she was living in 1956 when she applied for a SS number and whether she could sign her own name. Her parents' names - which it sounds like you know anyway - will be redacted, since June was born less than 100 years ago, if I remember correctly.
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Is her middle initial "N" ? If so, the people finder sites have her in the western states, not in Washington DC
veromi.net, which usually puts the most recent address first, says:
TUCSON, AZ
LAS VEGAS, NV 89142
HENDERSON, NV 89014
SAINT GEORGE, UT 84790
If you search familyseach.org for June Fox in Tucson she shows up, born 17 November 1928. They only have her in Tucson and Las Vegas.
If she was living in Tucson when she died in Washington DC, then
https://www.recorder.pima.gov/otherofficesmight be good place to start for a will, and maybe even a death record if it was recorded in her home state. It lists the various government offices in the county that handle the different records.
For DC deaths, it looks like:
https://dchealth.dc.gov/service/death-certificatesis the place to go.
If the links aren't the right government department, just google what you are looking for, that's all I did.
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by the way, Googling "June Fox" is a terrible experience. All I get are things that happened in June or were covered by Fox news. It reminds me of trying to find my wife's Page family. I wish there was a way to tell Google you were looking for a person, not a date, news outlet, or part of a book.