If you wanted honey, the easiest way to get it would have been to simply go to your own hives. While it's typically stated that keeping bees during the Early Medieval period was the preserve of members of religious communities - beehives were indeed popular in these contexts for the production of fine beeswax candles - the prerogative was far from exclusively theirs, and Domesday Book suggests that beehives were a fairly common occurrence in villages across 10th and 11th century England. The village of Syleham in the land of Robert of Tosny in Suffolk, for example, is listed as having a beehive, alongside its other livestock such as cobs, cattle and sheep. Interestingly, the beehive seems to have been a new addition since 1066. This is also the case in Ockenden, now in East London, where the addition of both a beehive and a substantial herd of cattle, significantly increased the value of the land from 1066 to 1086. Edgefield in Norfolk also gains a beehive post-Conquest, but already had a beehive prior to 1066 which suggests that they aren't a phenomenon which appears with the Normans. Yardley in Essex similarly has a pre-Conquest beehive, gaining a further 2 by 1066, while Playford, just outside Ipswich in Suffolk, had an impressive 6 beehives prior to 1066, which dwindled to only one by 1086, alongside a similar decline in livestock numbers.

Ranulf Peverel, who was tenant-in-chief of almost 100 settlements largely to the North-East of London, was clearly fond of honey; the number of beehives in his settlements, such as Billingsford and Springfield, proliferated rapidly post-1066, although perhaps not as much as his neighbour in Billingsford, Alan of Brittany, who augments the village's 5 pre-Conquest hives to 7 by 1086, alongside the 4 belonging to Peverel.

It's not entirely clear to what extent the honey would have been available to the general populace; presumably the hives themselves would have been started by enterprising villagers, given that the tenant-in-chief was often remote. The presence of honey in mandated food renders suggest that its portable nature may have rendered it particularly useful as a form of taxation, although rents are more likely to have been paid as service rather than necessarily in goods. An estate of 10 hides (very roughly 1,200 acres) was, according to the seventh century laws of Ine of Wessex, expected to produce some 10 vats (probably around 70kg) of honey in the form of food render. Ryan Lavelle's work on 'Farms of One Night' suggest that these food renders were specifically intended for royal vils, where the 'taxation' would have taken the form of providing food for King and his retainers and local witan as the court progressed on its continuous progression around the country. The British Beekeeping Association estimates that a typical hive produces around some 11kg of excess honey a year, so it is quite probable that a village like Billingsford with its 11 hives, would have been able to supply both its lord and the local population with honey. This is, of course, assuming that the Lord himself didn't have his own personal beehives. Indeed, evidence such as Lydgate's London Lickpenny suggest that the portable 'luxury' nature of honey and mead made it a handy commodity for farmers to sell at market to supplement their income.

The 10th Century Byzantine agricultural manual Geoponika has an array of advice for aspiring apiarists, such as:

The best hives, that is, containers for the swarms, are made from beechwood boards, or from fig, or equally from pine or Valonia oak; these should be one cubit wide and two cubits long, and rubbed on the outside with a kneaded mixture of ash and cow dung so that they are less likely to rot. They should be ventilated obliquely so that the wind, blowing gently, will dry and cool whatever is cobwebby and mouldy.

Of course, this construction means that beehives are unlikely to have left anything by way of archaeological record. We do, however, have a number of illustrations which also answer another part of your question: When it comes to actually acquiring bees in the first place, your medieval apiarist would have done much the same as his modern counterpart: find a handy wild nest or one that a neighbour needs removed, and take the bees to their own hive. Indeed, this illustration from a 12th Century English bestiary depicts an apiarist releasing captured bees from a sack into their new hive.