Clearing Event Shop Without CEs?

Item costs will be reduced so most likely yes
Also you can always roll for 3* event CE in FP gacha if you don't want to use SQ

Yes. This was one of the earliest FGO events and the difficulty is pretty low. If you cleared Camelot, you should be able to breeze through.

Some general tips:

Buy the cheaper event CEs from the shop ASAP and equip any Mooncell automation CEs you might have gotten from the FP gacha. You’ll get a free one after clearing the first main quest, so equip that ASAP too.

From there, just plan out what you want to acquire. You don’t have to Clear The Shop...just grab the stuff you need. I don’t need serpent jewels but I do need hearts, feathers, claws, and pages. 4* EXP is nice to have since I’m leveling/ascending servants to do all the interludes/rank up quests...I want pieces/monuments too. Your needs might be different.

The 3* Fire of Wisdom don’t give as much exp as you think, especially at higher levels, so those are low priority too. (Plus, EXP is easy to farm.)

Find a support (Waver) with the event CE and farm the 40ap node along the Gamepress guide recommendations. Events are the time to use apples, so use those if you’re not farming as quickly as you’d like.

Put some time into thinking about a quick farming setup. For the first 40 Ap node, Ushi + Arash + Waver should do it, especially if you have Ushi and Arash at np5 and max level. For the second node, borrow a good ST Lancer or bring Euryale/Robin with a Tamamo support. Arts chain your way to a full NP gague, buff with Fox’s Wedding, and have fun. (Or Lu Bu with Merlin support...but you’ll need some starting NP gague or a battery. Imaginary Element + Merlin’s 20% charge + Harry Potter costume will get you to 100% on your berserker, and then you can buff with Merlin and call it a day.)

I have a spreadsheet I've set up to project the event. With the assumptions:

1) You start with one mooncell from the FP gacha
2) You have a friend with a Moonlight fest CE (+3/stack)
3) You run the minimum # of quests to advance the main story
4) You buy 1 Moonlight Fest for dumplings, then 1 for rare dumplings as soon as you are able
5) You start with a full AP bar (I assumed L100 so 121 AP to start, but it turns out you can start with as little as 100, I think)
6) The drop rates are as published

Then you can farm 6006 dumplings and 1736 rare dumplings. This is enough to get the Claws from the rare shop, and from the regular shop: all of the 4* EXP, half of the Fous, all of the mats, half of the monuments and pieces. Of course, the final mix is a trade off between farming Expert vs Goddess, and the more rares you want, the less regulars you can get (the tradeoff is about 1:1).

So, no, you can't full clear without apples, but you can get a lot of the best stuff. At the end (with the presumption of +11/stack), you're at about 3.9 dumplings/AP (or if you want more rares, that's split about 60/40 in favor of the regulars), if you want to contemplate using apples to clear more.