So the answer was that the variable that contained the log information couldn't be combined with other strings in another variable.
$FullLog += $server + $log (would not work)
$FullLog += $log (would work)
Solution? I broke up the information:
foreach($server in $servers) {
$BeginDate=[System.Management.ManagementDateTimeConverter]::ToDMTFDateTime((get-date).AddDays(-30))
$mylog = Get-WmiObject Win32_NTLogEvent -filter "(logfile='system') AND (EventCode='19') AND (TimeWritten>'$BeginDate')" -computername $server
$First = $server + '= [{
"SuccessUpdate":"'
$Last = '"}]'
$First >> UpdateLog.js
$mylog >> UpdateLog.js
$Last >> UpdateLog.js
write-host $server "logs are uploaded."
}