• As @TobySpeight pointed out in a comment, the cat is redundant. Any time you have a line like cat "$file" | grep "$pattern" you can replace it with grep "$pattern" "$file"
• If your log entries are already ordered by time (at least for each user's login event), you can likely use tac instead of sort -r
• Can a user have logged in be their user ID? I assume not, but if they can, we have a bug where instead of sending their most recent login, you might send their most recent event of any kind (or something malformatted, might depend on how the rest of the log looks). Either way, if this turns out to be a problem it can easily be worked around by adding an anchor to the grep's pattern, or using grep -x
• The first two awk calls can be combined into a single awk -F' ' '!seen[$4]++ { p = index($2, "+"); print $1 "T" substr($2, 1, p-1) "Z", $4 }'
• By the way, awk looks dense without spaces, it'd be nicer if you added some
• Using awk and cut to remove [ and ] seems a bit awkward. Assuming those characters can't appear inside user IDs (since that's how user IDs are delimited it'd be annoying if they could) tr -d '[]' should do the job just fine
• Having a long line like that is annoying, but it becomes less annoying if it's split between multiple physical lines - which is possible
• Personally, I think it'd look better if you were to pipe your data into the while instead of sending it in from behind using < <(). It's a matter of opinion, but to me it better represents the flow of data and also subjectively looks nicer
• I imagine it might be useful to parse log files like this regardless of what they're called. Hardcoding the name file.log makes that harder. One alternative could be taking the name as a command line parameter and begin with something like grep -x "$TODAY.*logged in" "$1". But doing command line argument parsing properly is a pain, and pipes are pretty neat, so just operating on /dev/stdin instead could be an option - you'd have to call the script as ./submit-logins < file.log, which is slightly more annoying (especially if you aren't doing it from the command line), but it's a possibility

All in all, I'd probably recommend something like

#!/bin/bash

SERVER_API_URL="http://localhost:8080/api/user"

TODAY=$(date +'%Y-%m-%d')

grep -x "$TODAY.*logged in" /dev/stdin |
    tac |
    awk -F' ' '!seen[$4]++ { p = index($2, "+"); print $1 "T" substr($2, 1, p-1) "Z", $4 }' |
    tr -d '[]' |
    while read -r login_time user_id; do
        # following line is wrapped in echo for testing
        echo "curl -X PUT $SERVER_API_URL/$user_id/lastLoginTime/$login_time"
    done
```