Is there is an app that will easily allow our board members to access their webmail accounts from their smartphones?

TeamSideline mail services support POP and IMAP protocols, so webmail will work with any email client.  All smart phones have email clients built in.  For example, on an iPhone, you go to settings, Mail, Contacts, Calendars, and Add Account.

Checkout this article for more information:

https://teamsideline.zendesk.com/entries/20359281-How-to-setup-email-on-your-iPhone-iPad-or-IPod-Touch