Hello,
Yes this is possible, you will need to:
Create a new security role - eg. 'announcement access'
for each user that you wish to give access to the announcements module, apply the new security role to their user account
on the announcements module, go to the module settings, for the 'announcement access' security role tick edit permissions.
This will give those users access to add new announcements, but they will not be able to get to the admin functions of the module.
for further information on user accounts, security roles, etc. Have a look at:
I mentioned the ventrian subscription to you in a previous post for the subscription tools module, if you get a subscription from ventrian, then it is worth looking at their News Articles module as it provides a lot more functionality than the announcements module.