The Gmail users can set their gmail account to allow POP and or IMAP access. Meaning they can then use an email client to access their Gmail.
Gmail details what the settings are for many email clients. Filemaker works well with Outlook or Windows Mail on Windows - the email client needs to be set as the default email client. Mac Filemaker works well with Mail.app and maybe Thunderbird and MAYBE Outlook.
Hotmail users would have to use SMTP server for email sending. I am unsure if the hotmail SMTP servers will allow this. You have a difficult problem to solve.
Are you sure your clients need to send email from within Filemaker? Are they just using your database or are you selling them a database?
https://www.google.com/search?q=hotmail+smtp+filemaker Google "hotmail smtp filemaker"
Google "hotmail send email smtp site:forums.filemaker.com"
I created an eCatalog from which my designers can email their clients, images of items that they may be interested in. Some of the designers will be using FM Go so I'm assuming there will not be a problem sending email there. Others will be using the MAC or Windows Runtime solution for FM Pro which is where the problem is. I'm sorry I didn't explain my situation clearer on my first post. I am giving them my database to aid them in selling my products. Is there away to send these emails outside of FM? Would Save Records as PDF let me save their selections say to the desktop and then they can email from there?