There are two English nouns of this spelling, one meaning "a small spot or fleck", and one meaning "lard or fat". Since they do not share an origin, tonight's word is a double feature.
Speck, "a spot or fleck", is a native English word of uncertain origin. In Old English, the word is specca, and it is probably related to Dutch speckel. It may be a borrowing from Middle Dutch spekelen, "to sprinkle", which might also be the source of the English verb speckle, which is first attested in the 15th century. Spekke, the Middle English form of speck "a spot or fleck", may also be the source of that verb.
Speck, "lard or fat", is a more complicated case. Both the Oxford English Dictionary Online and etymonline.com list this word as originating in the 1630s (1633 first recorded use, according to the OEDO), and the OEDO lists it as a borrowing from Dutch spek and German Speck, both meaning "animal fat or lard". Etymonline.com, however, despite listing the origin date in the 1630s, identifies Old English spic, "bacon or lard", as the source of this word. The OEDO listing for that word, under spick, "lard or fat", has the word identified as obsolete, with its last recorded use in 1529, where it is spelled spycke. That word is also identified as having a common Germanic ancestor with Old Norse and Swedish cognates.
Now, while it is certainly possible that Old English spic fell out of use and was reborrowed a century later from a different Germanic source with only a very slight vowel change to distinguish it, I personally think there may be a simpler explanation. I believe speck "lard or fat" is simply a variant of spick "lard or fat".
The very slight vowel change in question is a lowering, from /ɪ/ to /ɛ/. These two vowels are very close, and in some dialects of English are interchangeable (look up the "pin/pen merger" for more information). English spelling was also not standardized in the late 16th and early 17th centuries, and vowels were the most variable elements in early English spelling. It is not difficult to imagine that spick, spycke, and speck are all the same word.
The word spick was common, as a cooking term, and was used often. Words like this are unlikely to suddenly fall off unless there is another word which just as suddenly rises to replace them. The two words which may have done this are lard and fat. Lard is attested from the 14th century, and is therefore unlikely to have pushed out spick. Fat is a native English word, but only gained the sense of "animal fat, lard" in the 1530s. It is therefore possible that fat replaced spick, but with lard already a synonym used specifically as a cooking term, it is unlikely that fat would have been used to push spick out of that particular lexical niche, especially since there is no indication of fat being used specifically as a cooking term the way spick and lard were. Therefore there is no clear reason why spick would have suddenly become obsolete, only for a very similar word with the exact same meaning to have been borrowed a mere century later.
It is not uncommon for there to be centuries-long gaps in the written record of attestations for any given word. A gap of 100 years in the written record is not necessarily an indication that a word has become obsolete, especially when the word in question is less likely to be attested in the written record at all. While there are some cookbooks and recipes that survive from this era of English, the majority of the surviving records we have are of a more erudite nature: music, philosophy, political writings, legal documents, poetry, etc. Words like spick/speck are unlikely to appear in such writings. Combined with the variable spelling of the time period, it is not unreasonable to assume that the word survived in the spoken record and was simply absent for a while from the written one.
Therefore, I believe that etymonline.com's suggested origin of Old English spic is in fact correct. If this is the case, however, their origin date of the 1630s is quite incorrect. If speck "lard or fat" is indeed a variation of Old English spic, then the word ought to be listed as native to English with an unknown Proto-Germanic source word, which was probably something like *spek (reconstruction my own).
https://www.etymonline.com/word/speck
https://www.oed.com/dictionary/speck_n4
https://www.oed.com/dictionary/spick_n1
https://www.etymonline.com/word/lard
https://www.etymonline.com/word/fat