Yes, you can have two 'and's here, but you are using more words than you need to and it can be rephrased to avoid the multiple 'and'. Here's a couple of examples:
- as GodNose says, you could simply use: '... by the faculty and staff'.
- use some punctuation: '... by the students, the teachers and the administrative staff'.
- you could also take out the multiple 'the': '... by all students, teachers and administrative staff'.