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Can any users other than administrators manage SpeakOut!

No, not as the plugin is.  But being able to specify roles that have access to SpeakOut! is actually a really good idea (thanks Simone A.), and I will look at making it an option in the security settings.

If you have some PHP coding experience there is a way around this, but the changes will be lost on any upgrade.  I am always deliberately vague with this stuff, if you don’t have the skills from what I tell you, you shouldn’t be doing it ;o)

open /wp-content/plugins/speakout/emailpetitions.php

Line 46 or thereabouts

if ( is_admin() ) {

is where it limits access

If you change that to

if ( is_admin() || current_user_can( 'editor' ) ) {

You would also allow editors to have access 

If you want multiple other roles it would look like

if ( is_admin() || current_user_can( 'editor' )|| current_user_can( 'author' ) ) {

But be careful to whom you give editing access i.e. probably not subscriber

I haven’t tested this, but there is no reason why it won’t work.