EDIT: Thanks to CQOzymandias for letting me know that this method has been patched and will no longer work after the V1.3 update.
There is an easter egg "cheat" that is mentioned in some other achievement solutions that can be re-purposed for this.
Every item in the game is linked to a specific item ID #. Anytime that name is referenced by a character in game (whether it is your name, your baby's name, or an animal name), you automatically get those items. In this case, you can use this to your advantage to complete any Help Wanted quest you obtain.
Spreadsheet with Item IDs: https://docs.google.com/spreadsheets/u/1/d/1pDWJ_ecANNex2CBG...
The potential limiting factor to this method would be money and progress. You need to have purchased either a coop or a barn in order to house animals and then have enough money to purchase a new animal anytime you need items you don't have. I've outlined the steps I followed below:
1. Obtain Help Wanted quest from board.
2. Head to Marnie's House [Top right area of the forest].
3. Look up the Item ID(s) of what you need for the quest. In this example I need a Fire Quartz (ID:82)
4. Speak to Marnie and purchase a chicken (or whatever animal suits your fancy) and make the name [Item ID#]
(my example would be a chicken named ).
5. Upon completion of the purchase, Marnie will send the animal to your farm and say it's name
, giving you the item(s) you needed.
6. Proceed to the person who needed the item and give it to them.
7. Rinse and repeat as many times as necessary.
If you need multiple items, you can string together multiple item IDs in the name.