Install SQL Server Express without administration rights
Unfortunately, no. You are installing a program and making service and registry changes, etc. You need to run setup as an account that has local administrative rights. No workaround to this that wouldn't involve some back door into administrative rights.

Best bet is to ask for someone to temporarily give you the rights or install it for you.

Source: The Microsoft SQL Server books online article for how to install SQL Server 2008 R2Express