Replication between SQL Server 2005 and Azure database
I'm not sure about the core of your question ("Can you set up replication between SQL Server 2005 and SQL Azure Database?"), but my assumption is a resounding "NO".

There are tools to migrate from SQL Server to Azure though - it's been discussed on Stack Overflow here and here.
The consensus seems to be that This SQL Azure Migration Wizard (CodePlex hosted project) is the way to go. It doesn't appear to be a live-sync migration though, so you will probably need to take an outage window unless I'm missing something...