Bandwidth doesn't matter for mining but Latency does. High lag can manifest in more rejects.
Ping your Avalon from your computer on the network over wifi. If your pings come back lower than 5ms (some of my miners on an optimized wifi bridge link get 1ms or less), you should be ok. If it's higher, a wired connection will benefit your setup.
Microsoft Windows XP [Version 5.1.2600]
(C) Copyright 1985-2001 Microsoft Corp.
C:\Documents and Settings\Owner>ping 192.168.1.11
Pinging 192.168.1.11 with 32 bytes of data:
Reply from 192.168.1.11: bytes=32 time=349ms TTL=64
Reply from 192.168.1.11: bytes=32 time=541ms TTL=64
Reply from 192.168.1.11: bytes=32 time=542ms TTL=64
Reply from 192.168.1.11: bytes=32 time=1ms TTL=64
Ping statistics for 192.168.1.11:
Packets: Sent = 4, Received = 4, Lost = 0 (0% loss),
Approximate round trip times in milli-seconds:
Minimum = 1ms, Maximum = 542ms, Average = 358ms
C:\Documents and Settings\Owner>
Edit: Second time///
C:\Documents and Settings\Owner>ping 192.168.1.11
Pinging 192.168.1.11 with 32 bytes of data:
Reply from 192.168.1.11: bytes=32 time=6ms TTL=64
Reply from 192.168.1.11: bytes=32 time=1ms TTL=64
Reply from 192.168.1.11: bytes=32 time=1ms TTL=64
Reply from 192.168.1.11: bytes=32 time=4ms TTL=64
Ping statistics for 192.168.1.11:
Packets: Sent = 4, Received = 4, Lost = 0 (0% loss),
Approximate round trip times in milli-seconds:
Minimum = 1ms, Maximum = 6ms, Average = 3ms
C:\Documents and Settings\Owner>
Third time...
C:\Documents and Settings\Owner>ping 192.168.1.11
Pinging 192.168.1.11 with 32 bytes of data:
Reply from 192.168.1.11: bytes=32 time=538ms TTL=64
Reply from 192.168.1.11: bytes=32 time=5ms TTL=64
Reply from 192.168.1.11: bytes=32 time=1ms TTL=64
Reply from 192.168.1.11: bytes=32 time=-534ms TTL=64
Ping statistics for 192.168.1.11:
Packets: Sent = 4, Received = 4, Lost = 0 (0% loss),
Approximate round trip times in milli-seconds:
Minimum = 1ms, Maximum = -534ms, Average = 2ms
C:\Documents and Settings\Owner>