It is possible to send an empty string when passing it as the argument to -o option:

$ zabbix_sender -z 127.0.0.1 -s 'Zabbix server' -k trapper.item -o '' -vv
zabbix_sender [23565]: DEBUG: answer [

{ "response":"success", "info":"processed: 1; failed: 0; total: 1; seconds spent: 0.000094"}

]
info from server: "processed: 1; failed: 0; total: 1; seconds spent: 0.000094"
sent: 1; skipped: 0; total: 1

However, when passing input from file it complains about invalid line syntax:

$ echo '"Zabbix server" trapper.item ""' | zabbix_sender -z 127.0.0.1 -i - -vv
zabbix_sender [5969]: Warning: [line 1] 'Key value' required
Sending failed.