As far as I know, the NPC that will give you the materials to start the local leve is usually indicated in your journal when you click on the corresponding leve.
Let's use Manine in Camp Bentbranch for example. Let's say you got this leve http://ffxiv.zam.com/en/localleve.html?ffxivlocalleve=120203
You would go see Manine, talk to him and he will give you the materials.. Once it says you got the mats from him, then you can select the synth in the requested items for A Mother's Carpentry...
It'll keep track of your synth progress and will let you know when you have enough items to hand in to finish the leve. You can keep going and use up all the mats they give, extra XP... Once you are done using up all your free mats, you go hand in the leve and get your reward.
If you were using your own logs, maybe that's why you can't hand it in. Use the provided logs and try then.