First post in this forum, so hopefully I'm doing this correctly.

If you are asking about the immediate claims after the collapse of the empires, then this is very complicated for the Western Roman Empire.The Byzantines are a bit easier since it was the Ottomans who finally sacked Constantinople and dominated the region, yet it is also historically complex.

As a sense of historic connection, the most fervent claims to the Roman and Byzantine Empires were those held by the Holy Roman Empire and the Russian Empire. Both of these developed out of religious lines with the Holy Roman Empire (through the Catholic Church) and the Russian Empire (through the Orthodox Church) having a general belief that they were destined to retake the position of political and religious importance held by the empires. The Holy Roman Empire (the collection of principalities that was united at first by Charlemagne in 800 that was formally broken up by Napoleon in 1806) held a belief that they were the descendants of power of the Western Roman Empire. The Holy Roman Empire (as the names suggests) held a significant connection to Roman power and influence. The emperors had varying degrees of power over their 1,000 year existence, but during much of the height of the Catholic Church political power, the popes were the source of that influence. On another note, the German eagle (which would be adopted by the Second Reich, the German Empire in the 1870s as a successor to the Holy Roman Empire--the first Reich) was a direct connection to the Roman eagle.

Similarly, the Russian Emperors adopted an eagle in their crest as a descendant of the Roman/Byzantine tradition. In the 19th century, the Russian Czars looked to the past of Slavic-Orthodox tradition in the Byzantine Empire to shape much of their foreign policy. Through World War I, the Czars looked to push into the Balkans, an aim largely prompted (at least from a propaganda/ideology perspective) to recapture the center of Orthodoxy, Constantinople, and act as the successor to Byzantine greatness. On another note the Russian use of the word "Czar" (and the German "Kaiser"), show another link to their historical connection to the Roman "Caesar".

Though this question is kind of... unanswerable, as the assumption of "legitimacy" is a political question, unsolvable and unenforceable except by consensus (which one does not exist), I would like to randomly toss the hat in for the Republic of Turkey.

They are the inheritors not only of the Ottoman Empire, which saw itself as the inheritors of the Roman Empire (and a bunch of other empires at the same time, including the seljuks and the caliphates), but also the Patriarch of Constantinople is still legally subject to the Republic of Turkey, a position previously reserved for the Byzantine emperors themselves.

But outside of that hat tossing, I would argue that there is no agreement upon who has the most legitimate claim, nor whether one exists at all.

For the western empire, in 476 the imperial insigna of Romulus Augustulus recognised as the last emperor (there was an "usurper" in Illyria) were sent to Constantinople, so technically the empire was reunified, and some barbarian chiefs showed signed of (theoretical) submission to the eastern emperor.

Do you mean who had the best claim to the entire Roman Empire when the Western Empire fell? Who had the best claim to the western roman empire when it fell? Who had the best claim to the Byzantine empire when it fell? or who had the best claim to the entire Roman empire when the Byzantine Empire fell?

"Roman/Byzantine Empire after it fell" is confusing. Based on your wording, you could be asking about two potential events, the 476AD sacking of Rome or the 1453 capture of Constantinople. With about a thousand years in between, there were indeed plenty of claims by kingdoms over the true inheritance of the Romans. To answer your question though, can you clarify which event you mean?