No, you will not be able to open an investment account with Dhan in case you do not have an Aadhaar Number.
As per regulations, Aadhaar is required mandatorily for following purpose -
Your PAN should be linked with your Aadhaar number
Aadhaar is required for eSigning of Account Opening Form (AOF)
Aadhaar is also required as a Proof of Address (PoA), however users have the option to submit Passport or Driving License for this purpose.