From memory, Rise of the Runelords manages to mostly avoid the bad design trap of 'you need a thief/wizard/cleric/str 20 guy to bypass this Locked Door and that is the only way to bypass it and the adventure cannot continue if you don't', so needing any specific class or character type is largely avoided (although any DM worth half a thimble of salt should be able to remove those whenever they show up).
Like most of the adventure paths, the designers occasionally throw in a far-too-hard encounter or a cakewalk, so you must eyeball them normally when running the campaign regardless - and alter them for your specific party.
An all-melee, no-healing party will naturally struggle with being kited by ranged foes, CC, and successive encounters. Of those, the last one is the only one that should be a major concern, as for story reasons it can be very hard to give them '8 days to heal up naturally' or whatever. Generally enemies with strong crowd control or kiting abilities should be weakened unless your party is made up of savvy players who will use strategems to increase their characters' combat effectiveness (like ambush, tripping flying archers into the floor and so on).
Player savviness can solve the 'no healing' problem, such as via Use Magic Device and Wands of Cure Light Wounds (and scrolls or potions for the rarer status ailments), adopting stealth tactics to not lose much hp in fights, buying piles of potions etc. They can also decide to bring along an npc with healing abilities which you can accentuate by placing interesting but not spotlight stealing npcs in their general path. But if these things don't happen or don't pan out you will need to adjust the strength of quickly successive encounters to accomodate the fact that the party has less or no magical healing, as well as providing windows for them to heal up naturally or visit a temple somewhere which the story has some of but probably not enough.
Note that there are a variety of ways for PCs to have npcs around doing jobs for them without it detracting from the players having agency or spotlight, and there's nothing wrong with the 'horde of hirelings' game style which is frankly far superior to someone being forced to play 'a healer' when they don't want to.
In any case, regardless of whether you have npcs or smart players or just 3 melee doofuses running at things, Rise of the Runelords is quite easily alterable to not require magical healing and be usable by nearly any party, required you have basic eyeballing encounter strength skills. If you run it purely RAW you will run into some issues or weird, immersion-breaking MMO like behaviour from the party that you won't be able to react to, but in general playing things purely RAW is not a fun time in any case so I don't recommend it.