Can I restart the holiday Mission?

I did the Egg event two days before I got my first mount and back then I didn't know that it gave me an item to go with it, I assumed that it would just be there in the list after I got my first mount.

Well it's Two days later and I finally got my Chokobo mount, and am only now realizing that the Egg isn't in my inventory. Not only do I not know what the item is called, in that two day time I have no idea if I moved or sold it unknowingly believing it to me a crafting item to junk.

Is there any way at all I can get a reset on the quest?

You cannot reset the quest if you already finished it. The item is called the Eggshilaration System Identification Key, and the icon looks like a key. It is unique/untradable so you get a warning if you try to sell/discard it. Perhaps, you already used the item to get the egg mount. If you open up your menu, and look under the character menu tab; you will see a menu tab called "Mount Guide." Open the guide up, and if you used the egg mount item, the egg mount should be listed in the mount guide. You can also shout to see if anybody has the quest active/mount key item in their inventory. If you ask them to link it for you; you can right click, and search to see if the item is located somewhere in your inventory.

I looked in my mounts and only my Chokobo is there, I don't have an achievement for the seasonal quest, did I miss something, because I went to talk to the lalafel the woman pointed me to.

Hatchtide video

Watch video and check to see if there is anything you missed.

If you do not have the achievement, then you did not completely finish the seasonal event. I would watch the video that Masekase posted; it might help you figure out the last thing you completed for the event.

Ohh my god I feel SO STUPID, I saw the giant Seasonal quest complete popup and just left.

Sounds like you resolved your problem.

You might find the following helpful in the future, though. If you can't find something, you can type

/isearch partofthename

and it'll give you a list of everything that you have anywhere that matches, which you can click for the locations of specific items. Some common item name parts for mount unlocks will be "horn", "whistle", and "key".

Yeah, oftentimes these events have a short series of quests that result in the final reward. In this case, its 3ish steps and then they hand you the key to your new egg(that is a weird sentence to type).