Suppose an individual is cited for a speeding infraction or served
with a civil subpoena/complaint and fails to show up to court because
at the same time the federal government deports the individual and as
a result the individual cannot show up to a state court/trial. Does
this individual have any recourse if a default judgement is entered
against them? Or otherwise avoiding penalties for not appearing in
court as a circumstance beyond their control occurred?
Yes and no.
There are a couple of sanctions for not showing up in court.
One sanction for failure to appear is being held in contempt of court for failure to appear. Inability to perform a court required action is a defense to contempt of court. So, if a physical appearance in required (as it is in criminal cases), deportation is a defense. Procedurally purging yourself of contempt of court sanctions might be cumbersome, but it would be possible in almost all instances if you acted within a reasonable time.
The other sanction for failure to appear is entry of a default judgment in a non-criminal case (some traffic cases are criminal in nature and carry the possibility of incarceration, some are civil in nature and carry the possibility of only a fine).
Default judgments can be set aside for "excusable neglect" (although often, only for a period of six months), which basically means, for circumstances beyond the control of the person charged with appearing or responding.
The case law of "excusable neglect" is very harsh with decisions of trial court judges finding that very hard circumstances still don't suffice to constitute excusable neglect being upheld on appeal given the trial judge's broad discretion in the matter. Most judges, most of time, however, are far more lenient than the appellate case law on the issue would suggest.
Federal cases on the topic are found in annotations to Federal Rules of Civil Procedure 6 and 60. Most states have analogous state rules of procedure with similar numbering, and even more states have very similar case law. Some key states, including California and New York, do not follow the federal system to number their civil procedure rules, and a handful of states would set substantively different legal standards for setting aside state court default judgments. The deadline for setting aside a default judgment is usually calculated from the date that the judgment is entered in the state court (assuming that there was valid service of process on the defendant, which otherwise raises a jurisdictional issue).
In a civil case, the issue is whether your deportation prevented you from filing a motion for extension of time to answer, or a document answering a complaint, usually by mail, or from hiring an attorney to do so. Judges have wide discretion in determining if conduct constitutes excusable neglect. During COVID-19, many traffic courts have permitted remote appearances via WebEx or a similar video-conferencing utility.
If you had advanced notice that a deportation was imminent and didn't make arrangements to respond to the civil complaint, or if the deportation to someplace with good internet access happened well in advance of the deadline for a response to a civil complaint, your failure to respond might not constitute excusable neglect, and the default judgment might stand.
On the other hand, if you were arrested to be deported en route to the courthouse, for a matter you had no reason to think was imminent (e.g. a seven year old criminal conviction), and then were held in immigration detention for a month while the deadline passed and you were deported, the grounds for setting aside a default judgment due to excusable neglect would be very solid, and a ruling to the contrary might even be overruled on appeal.