You'll want something that can either start a job via the command-line (so you can continue with your current process of kicking the tape run off after the Veeam job); or you'll need to be able to run your Veeam job via the command-line, in which case you'll need a tape job app that can run a command-line before it runs to tape. Backup Exec can do both (
bemcmd.exe is a hidden gem for command-line jobs, and the BE server supports pre-job commands).
In the UK, at least, Backup Exec is approx GB£500 for the main app. If you want to backup a second server, the Remote Agent is approx GB£200, provided you only want to backup files. If you want to backup a supported service (MSSQL, Oracle, VSphere, AD, etc), you'll need an agent for that service per machine (usually about GB£450). If you buy the service-specific agent, it usually contains an implicit Remote Agent, so you can backup files as well. The VSphere agent can backup VMs directly (i.e. without Veeam), but you'll need full-blown ESX (with VMware Consolidated Backup) to do it. Either way, most of the major third-party retailers offer the SKUs for the main app and the agents.
Of course, the easiest method is to have Veeam target a removable disk (e.g. USB or Firewire); you can take that offsite instead of a tape.