First, if you are talking about multiple applicants or signups at a time, as the plural possessive form you've adopted suggests, it makes sense to talk about multiple profile photos, since it's unlikely that your applicants all share the same photo:
Request access to your Xs' profile photos at any website.
Another issue here is that you're combining two nouns and separating them by a slash. That's usually frowned upon in written usage, either because it looks awkward (see he/she, described here as "clunky") or because choosing one of the two options would add clarity without much loss of meaning (what is the difference between an applicant and a signup?). The fact that the usage is discouraged means that applying other rules, such as the possessive, will lead into uncharted territory: there aren't many rules about applying specific rules to already-discouraged usages.
If you were to adjust the phrasing, choosing might look like this:
applicant/signups' -> applicants'
If you really can't choose, then rephrasing is often desirable. A minimal rephrasing involves rendering the slash as a conjunction like or:
applicant/signups' -> applicants' or signups'
Once the construction is expanded into a list of two nouns, we can apply rules as explained in this Stack Exchange answer or this post: each noun owns a separate profile photo, so both nouns would take the possessive:
Request access to your applicants' or signups' profile photos at any website.
Let's say you're committed to the slash, conventions be damned, but you want the possessive to work. Which form you pick depends on how you treat the resulting construction.
Is applicant/signup a noun phrase formed with a slash, as suggested by a user in this Reddit post? Only put the plural possessive at the end, as you would do with another noun phrase: applicant/signups'.
Is the slash an abbreviation for or, as suggested in this resource? Apply the plural possessive to both: applicants'/signups'.
I favor 2 for the sake of consistency with an occasional pronominal usage: he/she usually converts to his/her, not he/her. That seems like the least awkward option:
Request access to your applicants'/signups' profile photos at any website.