Previous poster is right; if this is for a girl and you are using accents, then it needs to be Esmee (with accent on the first e). Esme (accent) with just one e is very much a French male name.
For those saying 'Esmay' is wrong, well, as other posters have said, it's as near as many English-speakes can get to the French original, which is, as others have also said, very roughly Esmeh. Listen here: forvo.com/word/esm%C3%A9/#fr
For those Scots like myself, the Scottish accent has vowel sounds that are often very close to French and just don't exist in modern English.
Esme/Esmee as a personal name originated in Scotland. The name is not "Es-maaay" but it is Es-meh or Es-mey - the same sound as in 'James' (= Scots James/Jehms). Or, if it helps, just imagine that you are in Dundee and asking for a pie - it's the same sound in the Dundonian accent as 'news -peh-per'. And I KNOW - my granny was from there.
The accent is on the FIRST syllable.
Brilliant discussion of east coast Scots vowels by an Italian - -she's really good
Personally, I much prefer (for a girl) the French Edmee (acute accent on first syllable). It means much the same as Esme/Esmee. It's pronounced Ed-mey. forvo.com/word/edm%C3%A9e/#fr