The phrase “ye olde” is a great example of how marketing language successful and get stuck into the public consciousness, and become a part of speech despite being linguistically incorrect. Or more accurately, in this case, a-historical.

The phrase” ye olde shoppe” or “ye olde steak house” is not Old English or even Middle English. It was invented in the late 1800s to sound old timey in order to sell products.
The word old was not commonly spelled “olde” during the Middle English period. It was more often rendered as “auld” or “all” or “ould”, although English didn’t have standardised spelling at the time. So it is not beyond the realm of possibility that some speakers spelt the word “olde”.
What is more problematic is the letter Y in the word “ye”. This construction derives from Old and Middle English but it was not pronounced like a Modern English Y sound. It was actually a TH sound, rendering the word “ye” the orthographic forebear of the word “the”.

The transposition of “ye” and “the” is due to the fact that early printing presses were made in Germany and had movable type in with contemporary German letters. Thus their script was only capable of handling English letters that were the same as in German, which meant certain English letters had to be recalibrated, including a runic letter called the thorn. Words that had previously been spelled with thorn were realized with the letter Y in printed materials, which within a century led to the loss of the letter thorn.
The digraph TH had been previously used in Latin to transliterate Greek loan words, and later in Old English to represent roughly the sounds it represents today. By the 9th century, the runic letter thorn had replaced the digraph TH, but they swapped places again in the 16th century, due to the influences of printing technology.
So yeah “ye olde shoppe” is just pseudo-archaic marketing language from the late 1800s that was picked from some 15th century Middle English texts. It is constructed from a mix of different orthographies, and should just be pronounced The Old Shop. But if it gives you pleasure to say Yee Oldie Shoppy, then have at it! Language is a fluid construct driven by speakers, so anything goes really.