Using Slack integration, can I create a channel that can be automatically populated by new ideas raised on our Salesforce community?

It should be possible. You need to customize your code accordingly and query for user raised idea. Here is a sample apex class that posts to slack. Please make changes according to your requirement

public with sharing class SlackOpportunityPublisher {
    private static final String slackURL = 'YOUR_WEBHOOK_URL';
    public class Oppty {
        @InvocableVariable(label='Opportunity Name')
        public String opptyName;
        @InvocableVariable(label='Stage')
        public String stage;
    }
    @InvocableMethod(label='Post to Slack')
    public static void postToSlack(List<Oppty> oppties) {
        Oppty o = oppties[0]; // If bulk, only post first to avoid overloading Slack channel
        Map<String,Object> msg = new Map<String,Object>();
        msg.put('text', 'The following opportunity has changed:\n' + o.opptyName + '\nNew Stage: *' + o.stage + '*');
        msg.put('mrkdwn', true);
        String body = JSON.serialize(msg);    
        System.enqueueJob(new QueueableSlackCall(slackURL, 'POST', body));
    }
    public class QueueableSlackCall implements System.Queueable, Database.AllowsCallouts {
        private final String url;
        private final String method;
        private final String body;
        public QueueableSlackCall(String url, String method, String body) {
            this.url = url;
            this.method = method;
            this.body = body;
        }
        public void execute(System.QueueableContext ctx) {
            HttpRequest req = new HttpRequest();
            req.setEndpoint(url);
            req.setMethod(method);
            req.setBody(body);
            Http http = new Http();
            HttpResponse res = http.send(req);
        }
    }
}