Joseph O'Callaghan addresses this question straight away in Reconquest and Crusade in Medieval Spain. For some background his basic thesis in the book is that the wars between (roughly) Christians and Muslims in Spain are one war, stretching from the 8th century to 1500. This ongoing struggle is "transformed" by crusader ideology, by the papacy, in the 12C-13C.

O'Callagahan argues that reconquest as a concept is not an anachronism (unfortunately my notes don't say anything about the use of the term "reconquista" in Spanish-language sources). As early as the 8th century, Christian chronicles speak of the "salvation of Spain," casting the war as an inevitable coming struggle with victory at the end. Chroniclers and diplomats worked hard to keep the idea of the coming reconquest alive even when it didn't seem like a possibility. Medieval historians invented continuities between the Visigothic kingdom and the early Iberian Latin-Christian kingdoms in the north (Asturias). Iberian Christian kings drew up treaties to split up future conquests. O'Callaghan notes that Christians really did see the Muslims as invaders to be swept away: their goal was to kick Muslims out, not convert them. Medieval Spanish Christians did understand their efforts against al-Andalus as a reconquest.

I agree with /u/sunagainstgold 's (and O'Callaghan's) assertions that the concept of reconquest was known to Medieval Iberians and is useful to describe their campaigns; and would like to add a few points on the usage of the term 'reconquista'.

Do any contemporary sources mention a "manifest destiny" like drive to reconquer the peninsula?

The 'Primera Crónica General' was written in the late 13th c., started under predecessors of Alfonso X. of Castile and continued under Alfonso. It was a project involving various scribes and translators, but due to the king's participation can be seen as expressing royal viewpoints that proved influential on later universal chronicles. The providential conception of history already mentioned is evident, e.g. when the Visigoths' defeat is portrayed as divine punishment for their kings' sins, while the “survival” of the Northern Christian realms is also seen as divine intervention in order to upkeep the “light of Christianity” in Iberia. The description of how God helped the first Asturian ruler at the mythical battle of Covadonga (in the 8th c.) can be seen as an important legitimating strategy for Christian conquests, linking Visigothic territories with the later Christian kingdoms. Formulations used in the chronicle include praising Spain as a “paraíso de Dios” (paradise of God) and most elevated of all countries, but no clear mention of the term reconquista is made. I took the 'Primera Crónica General' as an example of Medieval chronicles and now turn to writings written following the end of Muslim rule in Iberia.

If the term reconquista wasn't yet used at the time, as the example suggests, when did it come into regular use? Martín Ríos Saloma wrote an article tracing this question, where he argues that the term itself goes back only to the 18th century (in 'De la Restauración a la Reconquista: la construcción de un mito nacional - Una revisión historiográfica. Siglos XVI-XIX, in En la España Medieval 28, 2005). He looks at the concept's development between the 16th and 19th cs. in quite some detail. His main points go as follows: During the 16th and 17th cs. the focus on historiographical works on the Iberian Christian campaigns lay on the supposed “restauration” of Visigothic rule by the later kingdoms, rather than on these kingdoms' military conquests. Thus authors would talk of the “perdida y restauración de España”, culminating in the expulsion of Muslims in 1492 according to them. Saloma identifies the first use of reconquista in the 'Compendio cronológico de la historia de España' (1795-1803) by José Ortiz y Sanz. I'll only quote the example itself for brevity's sake: „the desesperation, the sorrow of seeing their homeland lost, and above all, the religion and favours of the heavens encouraged them to think not only of their own defense, but also of the reconquest (en reconquistar) the homeland from the enemy's hands [...]“ (my translation). Ortiz could use the term in this way, referring to the Spanish „homeland“ (non-existent in the 14. and 15. c.) and to „enemies of the faith“ because he was building on the works of earlier authors who had started linking the concept with patriotic messages, as well as using words like „conquer“ in this context.

In the article's last part Saloma goes on to describe the further adaptation of the reconquista in romantic and nationalist writings of the 19th c. For him, the term became only more current in the 1840's while the formerly more common „restauración“ stayed in use at least until the first half of the 19th c. The stronger use of the term reconquista in this way meant a change in focus from the „reestablishment“ of the Spanish monarchy and its institutions to a stronger emphasis on the Christian military campaigns – casting these campaigns as as a national founding myth. For Saloma this change in emphasis was connected to 19th c. developments in Spain like the consolidation of the liberal citizenry and a growing necessity to distinguish historically between Spaniards and other nations, by highlighting the „victory“ over Spanish Muslims. As I said, I think that the concept reconquista is nonetheless useful to describe these developments in Medieval Iberia, as described by Castilians like Alfonso X. themselves – while keeping these roots of the term in later nationalist Spanish thought in mind.

The concept indeed was not inconceivable but there's not one single answer on whether it was considered the manifest destiny. As you know, there were different periods in what we now call the Reconquista and lots of different actors so there's no simple answer that would be valid for every period and kingdom.

The one-single-war narrative as outlined by u/sunagainstgold was actually a peculiar concept of the one Christian kingdom that eventually came to dominate Spain, that of Castile. It was founded by Visigothic refugees from the south in the mountains of Asturias shortly after the Muslim conquest (or at least by people who their descendants later claimed had been Visigothic noblemen from the south, we don't really know and there are good reasons to believe it was not true). The first Asturian kings soon started basing their legitimacy on claiming to be basically the successor state to the Visigothic kingdom of Spain that had been destroyed by the Arab conquest. When they built a new capital at Oviedo they tried to emulate the geography of Toledo, the lost Visigoth capital (even though the site for the new city didn't look much like Toledo, frankly). They undertook a vast building program, building many churches in a peculiar style that was more late antiquity than anything the Visigoths had really built at Toledo or elsewhere. Oh, and they had the local church at their side. The church writers - most notably, Beatus of Liébana - naturally saw the Muslim conquest of Spain as a sign of the impending Apocalypse. It was now the work of the Asturian kings to retake the land from the infidel and restore the lost Visigothic kingdom. Conveniently, it is at the same time that the supposed grave of St James the Apostle, Santiago, is 'found' in Galicia. In a couple of centuries Santiago becomes the patron saint of the Reconquista and of the Asturian (now Leonese) kingdom. Quite a lot of battles are won only because the Apostle himself appears at some crucial moment to save the day and lead Christians to victory - or that's what the early chronicles, compiled in Asturian monasteries would want us to believe.

The best part of it? For these early centuries we have no evidence that there was any reconquest at all. There may have been a couple of insignificant skirmishes and the Christians did mount a raid or three deep into Muslim territory and come back with plunder and Christians. But what early expansion there was actually seems to have consisted in Asturian kings taking over no man's land that had been abandoned by Muslims and resettling it with those Christians they brought back from Muslim lands. At some point they would grow insolent enough for Muslims to mount a punitive expedition. Most famously, Al Mansur campaigned in the north a lot, at some point managing to sack both Barcelona and Compostela, the two cities lying at the opposite ends of the North. But the spin that the church and the Asturian kings put on the events later came to be the accepted narrative of the Reconquista. For an early 'official' version, see the comment by u/drylaw above. Note that the Primera Crónica General was composed after the bulk of the Reconquest already happened in the 13th century. The Reconquista was already a thing by then.

But Asturias was not the only early nucleus of Christian resistance to Muslims. Other Christian polities emerged in the northern mountains, including Pamplona, Aragon, Sobrarbe, Ribagorza and various counties in what is now Catalonia. Now, Old Catalonia was conquered from the Muslims by the Carolingians (and the local faction, as opposed to Franks, were known as 'Goths'). But the other kingdoms and counties claimed no Visigothic descent that their legitimacy could be based on. And, frankly, them having this single overarching narrative of a constant struggle against the infidel the way Asturias did just does not seem very consistent with what we know about their early history. I recently wrote a lengthy [comment] (https://www.reddit.com/r/AskHistorians/comments/4iu14t/why_didnt_the_basques_conquer_south_during_the/d32a995) on the early Pamplona that you may find interesting as an example of what challenges they faced in the upper Ebro valley against the Muslims. Granted, after the collapse of the Cordoba Caliphate in 1031 made actual conquest possible they all indulged in some expansion to the south. Better yet, Pamplona actually took Nájera from the Moors as early as 924, possibly the first Christian state to conquer anywhere by force in the Reconquista (and to do this they did not need any of this faux Visigothic posturing the Asturians were so fond of). But even after 1031 some Christian states were as keen to expand at the expense of fellow Christians as they were in fighting the Muslims. If not more, actually. Case in point, the rulers of Catalonia (and Aragon, post-union) were much more interested in expansion north, across the Pyrenees. It was only after king Pedro II got killed at Muret in 1213 that they re-oriented towards conquest in the south.

To borrow a turn of phrase from Charles Julian Bishko, the Reconquista was initially an 'Iberian civil war' that later got highjacked, if you will, and cast in religious terms, becoming a 'Spanish crusade'. This was mainly due to influence from outside of the peninsula (Cluny and Rome, mainly) and the first instance that we know of dates back to 1064 and the attack on Barbastro in Aragon. Over the course of the 13th century, when it became the accepted narrative, most of Spain was actually reconquered.