Where can I find original manuscripts of the songs written by the
French composer Marguerite Monnot? She wrote many songs for Edith
Piaf.

Some manuscripts by Marguerite Monnot are mentioned here:
http://www.sqrm.qc.ca/Archives/Bnq/Bibliotheque_Nationale_du_Quebec.htm

La Société québécoise de recherche en musique (SQRM):
http://www.sqrm.qc.ca/accueil.htm

I can't seem to search this web site. How did you get to Monnot?

According to (as mentioned in my previous comment)
http://www.sqrm.qc.ca/Archives/Bnq/Bibliotheque_Nationale_du_Quebec.htm

the collection of the Bibliothèque Nationale du Québec has "partitions
diverses de compositeurs et paroliers dont Dantès Belleau, Alexis
Contant, Michel J. Dussault, Achille Fortier, Arthur Letondal,  
Marguerite Monnot  , Georges Panneton" ("11 cm de documents
textuels").

For further details you could ask the SQRM
http://www.sqrm.qc.ca/contact.htm

or the Bibliothèque nationale du Québec
http://www.bnquebec.ca/en/coordonnees_en/coordonnees_en.htm

Thank you. Just one more question: can you find out if the
Bibliotheque Nationale in Paris, France, also has some Monnot
manuscripts and whether they are available for research?

The BnF has more than one catalogue for you to browse:
http://www.bnf.fr/pages/zNavigat/frame/catalog.htm

However, not all of the holdings are searchable online ("Une partie
des collections de la BnF n'est pas encore recensée dans les
catalogues informatisés"):
http://www.bnf.fr/pages/zNavigat/frame/catalog.htm (then click on
"Autres catalogues").

BN-OPALINE:
"La base BN-OPALINE a été conçue et créée à la Bibliothèque nationale
de France dans le but d'informatiser les fichiers et catalogues des
départements conservant des documents " spécialisés " (...  Musique):
http://www.bnf.fr/pages/catalog/opaline.htm#1

Using BN-OPALINE http://opaline.bnf.fr/ 
and then clicking on "Musique" you can enter "Monnot" as "auteur".

This should lead you to 122 results for "Monnot, Marguerite, 1903-1961".
I am not quite sure if these results are referring to manuscripts or not. 

If you would like to write to the BnF you can use the following form:
http://www.bnf.fr/pages/zNavigat/frame/outils.htm?ancre=courriel.htm

If one of the links does not work do let me know.